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Description  |
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TECHNICAL FIELD
This invention relates to component useful in optical computing circuits,
and more particularly to a saturated optical interaction gate useful in
all-optical computing circuits.
BACKGROUND ART
U.S. Pat. No. 4,811,258 discloses a reversible all optical implementation
of an interaction gate. One embodiment of the interaction gate disclosed
was a dual beam version of an optical nonlinear interface. Another
embodiment was a dual beam Fabry-Perot.
A nonlinear interface is commonly known as a plane interface between two
dielectric media, one of which has an intensity dependent nonlinear
refractive index. There has been much interest in the nonlinear interface
(NI) and its behavior over the past decade (see A. E. Kaplan, "Theory of
Hysteresis Reflection and Refraction of Light by a Boundary of a Nonlinear
Medium", Sov. Phys. JETP 45, 896 (1977); P. W. Smith, W. J. Tomlinson, P.
J. Maloney and J.-P. Hermann, "Experimental Studies of a Nonlinear
Interface", IEEE QE-17, 340 (1981); W. J. Tomlinson, J. P. Gordon, P. W.
Smith and A. E. Kaplan, "Reflection of a Gaussian Beam at a Nonlinear
Interface", Appl. Opt. 21, 2041 (1982); P. W. Smith and W. J. Tomlinson,
"Nonlinear Optical Interfaces: Switching Behavior", IEEE QE-20, 30 (1984);
R. Cuykendall and D. Andersen, "Reversible Computing: All-Optical
Implementation of Interaction and Priese Gates", Opt. Comm. 62, 232
(1987); "Reversible Optical Computing Circuits", Opt. Lett. 12, 542
(1987); R. Cuykendall, "Three-Port Reversible Logic", Appl. Opt. 27, 1772
(1988); D. R. Andersen, R. Cuykendall and J. Regan, "Slam-Vectorized
Calculation of Refraction and Reflection for a Gaussian Beam at a
Nonlinear Interface in the Presence of a Diffusive Kerr-like
Nonlinearity", Comp. Phys. Commun. 48, 255 (1988); and R. Cuykendall and
K. Strobl, "Thin Film Computing with the Nonlinear Interface", JOSA-B, to
be published, (1989). This was due primarily to its potential for
ultrafast (subpicosecond) switching. Early experiments and simulations
both indicated intrinsic limitations to beam and switching quality, i.e.
transmitted beam breakup associated with multiple intensity thresholds and
low contrast between total internal reflection and transmission.
Those concerned with these and other problems recognize the need for an
interaction gate with improved switching quality.
DISCLOSURE OF THE INVENTION
The present invention provides a saturated optical interaction gate based
on the interaction gate disclosed in U.S. Pat. No. 4,811,258, which Patent
is hereby incorporated herein by reference. The saturated optical
interaction gate includes a nonlinear interface formed of materials chosen
to enhance reflection of beams at a first total intensity and to enhance
transmission of the beams at a second total intensity.
An object of the present invention is the provision of an improved
saturated optical interaction gate.
Another object is to provide a saturated optical interaction gate having
enhanced switching qualities.
A further object of the invention is the provision of a saturated optical
gate that is utilized in designing all-optical circuits.
An additional object is to provide an optical interaction gate that can be
designed alternatively to transmit either at high or at low intensity.
These and other attributes of the invention will be become clear upon a
thorough study of the following description of the best mode for carrying
out the invention, particularly when reviewed in conjunction with the
drawings.
BRIEF DESCRIPTION OF THE DRAWING
FIG. 1 is a schematic showing the experimental set up.
FIG. 2 is a graph showing the saturation enhanced NI switching for an
incident glancing angle .PSI.=7.3.degree. and an artificial Kerr medium
having a volume concentration of 10.5%.
FIG. 3 is a graph showing the intensity dependence of NI reflectivity for a
volume concentration of 7.7% at .PSI.=7.3.degree.: (a) best results
obtained for horizontally oriented NI; and (b) typical measurement for a
vertical NI with softer saturation.
FIG. 4 is a schematic diagram of the nonlinear interface configuration and
coordinate systems used for the computations. Computation starts in the
negative Z halfspace and progresses versus the positive Z half-space. The
linear medium fills the positive X half-space and nonlinear the negative X
half-space.
FIG. 5 is a typical `saw-tooth` reflectivity curve predicted by the
standard model simulating the reflection and transmission of a
two-dimensional Gaussian beam incident on the NI having a Kerr-like
nonlinearity for the refractive index. Interface and beam parameters are
.DELTA.=0.01, n.sub.o =1.391, .PSI.=7.3.degree. and w.sub.o =10.lambda.=10
.mu.m.
FIG. 6 is a graph illustrating the intensity distribution for the reflected
and transmitted beam calculated with the standard NI model for the
parameter used to generate FIG. 5 at an incident intensity n.sub.2
I.sub.max /.DELTA.=11.6. Roughly 80% of the incident beam gets transmitted
in the nonlinear medium and splits up into 13 self-focused channels.
FIG. 7 is a graph depicting the hard limited SNI model prediction with (a)
`ideal` saturation .DELTA..sub.sat =3.DELTA. (big squares connected by
thick lines) and (b) slight overshooting .DELTA..sub.sat =1.5.DELTA.
(small squares with thin line). Note the large difference in switching
behavior for small deviation from the ideal saturation case.
FIG. 8 is a graph showing the hard limited SNI model prediction as in FIG.
7 with (a) .DELTA..sub.sat =3.DELTA. (squares connected by thin lines) and
(b) .DELTA..sub.sat =.infin. (dotted line). Note the small difference in
switching behavior when going from case (a) to case (b).
FIG. 9 is a graph illustrating the beam profile predicted for the hard
limited SNI corresponding to FIG. 10(a): (a) n.sub.2 I.sub.max
/.DELTA.=11.6 and (b) n.sub.2 I.sub.max /.DELTA.=2.64.
FIG. 10 is a graph showing the calculated NI reflectivity predicted by the
hard limited SNI model for .DELTA..sub.sat =.DELTA. as a function of
normalized intensity for various glancing angles: (a) .PSI.=3.80.degree.
(triangle); (b) .PSI.=4.14.degree. (diamonds), (c) .PSI.=4.5.degree.
(spheres) and (d) .PSI.=5.0.degree. (squares). Interface and beam
parameters as in FIG. 5.
FIGS. 11(a)-11(b) are graphs showing the beam profile predicted for the
hard limited SNI corresponding to FIG. 10(d): (a) n.sub.2 I.sub.max
/.DELTA.=4.41 and (b) n.sub.2 I.sub.max /.DELTA.=11.6.
FIG. 12 shows the calculated NI reflectivity predicted by the hard limited
SNI model for .DELTA..sub.sat =.DELTA. as a function of normalized
intensity at .PSI.=3.8. The different symbols represent different beam
waists w.sub.o. Interface and beam parameters as in FIG. 5.
FIG. 13 is a graph showing the beam waist dependence off the SNI intensity
threshold for the curves shown in FIG. 12. The solid line represent the
best fit to a power dependence (y=1.55 x.sup.-1.95)
FIGS. 14(a)-14(b) are graphs depicting the SNI model beam profile
corresponding to FIG. 12 with w.sub.o =5.lambda.: (a) n.sub.2 I.sub.max.
/.DELTA.=6.3 and (b) n.sub.2 I.sub.max. =7.9. Note that the increase in
lateral and angle shift and beam waist change for the reflected beam when
going from case (a) to (b). The latter is right at the threshold to
partially transmission.
FIG. 15 is a graph showing the measured intensity dependence on NI
reflectivity for n.sub.o =1.391 and the calculated values .DELTA.=0.031,
n.sub.2 =0.56 cm.sup.2 /MW and w.sub.o =4.4 .mu.m=8.6.lambda..
FIGS. 16(a)-16(b) are schematic diagrams showing (a) the nonlinear
interface, the corresponding material selection table and (b) the
Fabry-Perot embodiment with its material selection table.
BEST MODE FOR CARRYING OUT THE INVENTION
An experiment (see K. H. Strobl and R. Cuykendall, submitted to Phys. Rev.
A (1989)) investigating reflectivity of a light beam crossing a plane
interface between two dielectric materials with a low intensity refractive
index offset .DELTA., one having an intensity dependent index, showed
unexpected plane wave-like behavior (see A. E. Kaplan, JETP Lett. 24, 114
(1976); Sov. Phys. JETP 45, 896 (1977)) [high contrast single-threshold
switching]. The reasons for such behavior were not evident from the
particular findings reported. However, saturation of the nonlinear index
was suspected (see K. H. Strobl and R. Cuykendall, submitted to Phys. Rev.
A (1989)) as a possible mechanism washing out the undesirable multiple
thresholds predicted by the standard two-dimensional simulation (see W. J.
Tomlinson, J. P. Gordon, P. W. Smith and A. E. Kaplan, Appl. Opt. 21, 2041
(1982); P. W. Smith and W. J. Tomlinson, IEEE QE-20, 30 (1984)) of a
Gaussian beam incident at such a nonlinear interface (NI). Moreover, the
observation of multiple intensity thresholds (see P. W. Smith and W. J.
Tomlinson, IEEE QE-20, 30 (1984)) led to the conclusion they were
characteristic of NI behavior. Recent theoretical studies (see R.
Cuykendall, Appl. Opt. 27, 1772 (1988); R. Cuykendall and K. H. Strobl, in
press, JOSA-B (May, 1989)) have shown that improved switching behavior
[reduction in beam breakup due to multiple thresholds] can be achieved by
including diffusion in the two-dimensional model. The simulation of a
saturated nonlinear interface (SNI) confirms that saturation effects can
also cause the observed plane wave-like behavior. Experimental results of
enhanced NI switching provide evidence that saturation of the non-linear
refractive index is the primary influence.
The standard two-dimensional Gaussian model prediction (see K. H. Strobl
and R. Cuykendall, submitted to Phys. Rev. A (1989); W. J. Tomlinson, J.
P. Gordon, P. W. Smith and A. E. Kaplan, Appl. Opt. 21, 2041 (1982); P. W.
Smith and W. J. Tomlinson, IEEE QE-20, 30 (1984)) for an NI is based on
the assumption that the induced index change .DELTA.n is directly
proportional to the local light intensity I, i.e.
.DELTA.n=n.sub.2 I. (1)
The Kerr coefficient n.sub.2 is a measure of the strength of the nonlinear
response. Under this condition the model predicts that each interference
fringe [formed by the incident and reflected wave at the interface]
suddenly switches through the interface when its peak intensity reaches a
threshold value, resulting in a sharp drop in total reflectivity and
formation of an additional self-focused channel propagating in the
nonlinear medium. Between adjacent intensity thresholds the reflectivity
increases with intensity due to increased index mismatch for the already
transmitted fringes. By saturating equation (1) in such a way that
transmitted fringes can only cause limited overshooting of the ideal index
change .DELTA., reflectivity increase between thresholds should be
preventable. Since the reflectivity can then only decrease, deeper total
switching would result with a pronounced initial drop. Depending on the
amount of saturation, subsequent reflectivity `oscillations` will be
lessened or possibly nonexistent. For this reason better switching quality
is expected for a saturated NI than for a standard NI. This SNI picture
can be tested experimentally if a way can be found to influence the amount
of saturation in the nonlinear medium.
The experimental set up used is shown in FIG. 1. Briefly, a p-polarized
output of a CW argon ion laser (.lambda.=514.5 nm) was focused at the
horizontal interface to a spot size with a theoretical 1/e amplitude
radius w.sub.o and a far-field diffraction angle .PSI..sub.D in the linear
medium. A polarizer combined with a rotatable .lambda./2-plate allowed
continuous attenuation of the incident light beam while identical power
meters (I and R) measured the incident and reflected power each second.
The refractive index for the linear medium (LiF) is n.sub.o =1.391. The
nonlinear medium consisted of an aqueous suspension of highly uniform
polystyrene spheres with a radius r.apprxeq..lambda./4. In addition, the
nonlinear liquid contained small amounts of surfactant and was saturated
with LiF in order to prevent surface etching of the LiF crystal.
A simple explanation of the behavior of dielectric spheres in suspensions
exposed to an electromagnetic field gradient can be given (see P. W. Smith
and W. J. Tomlinson, IEEE QE-20, 30 (1984); P. W. Smith, P. J. Maloney and
A. Ashkin, Opt. Lett. 7, 347 (1982)). The light-induced force on the
spheres is proportional to the gradient of a field, and therefore can be
expressed in terms of a potential .phi.. This force attracts the sphere in
the high intensity region, increasing the particle density locally,
resulting in a refractive index change, and is opposed by thermal
diffusion (Brownian motion) of the particles. The quasi-stationary
equilibrium sphere-density distribution caused by the potential .phi. can
be found from a standard diffusion model in which the sphere-density is
proportional to exp-(.phi./kT). Assuming the validity of this explanation
it was found that for a not-too-intense light beam (.phi..sub.max
.ltoreq.kT) the induced density distribution is only negligibly sharper
than the intensity profile. Under such conditions the thermal diffusion
counterbalances the optical pressure, resulting in a density distribution
which follows the intensity variation. The effect of Brownian diffusion in
this particular system is thus to simulate a diffusionless Kerr-like
nonlinearity, i.e. equation (1). Based on this description, it can be
shown (see P. W. Smith, P. J. Maloney and A. Ashkin, Opt. Lett. 7, 347
(1982)) that for spheres with a radius r much smaller than the wavelength
(Rayleigh regime),
##EQU1##
where n.sub.l is the refractive index of the surrounding liquid, n.sub.s,l
is the ratio between the refractive indices for the spheres and the
liquid, and N is the number of spheres per unit volume. The sphere size
was selected to give the highest possible nonlinearity while still
approximately satisfying the Rayleigh condition. This allowed the best
chance to observe saturation effects for the available CW laser power
(.apprxeq.1.3 W). Since the trade off for suoh a high nonlinearity was a
nonlinear medium with a very slow response time (.apprxeq.100 ms), only
quasistationary NI behavior was investigated by using CW operation
together with a slow intensity attenuation (111 seconds from high to low).
Equation (1) neglects, among other effects, the actual sphere size, the
agglomeration and the coulomb repulsion/attraction due to surface
charges/induced dipole moment and dissolved ions. However, it is clear
that the spheres cannot be packed denser than hexagonal or cubic close
packing permits. This gives an upper limit of .DELTA.n.sub.max. =0.14 for
the induced index change in the polystyrene water suspension. Long before
that limit is reached, coulomb forces and particle interactions resulting
in increased viscosity will slow down the response by saturating the
nonlinear index, resulting in an effective refractive index change which
is smaller than the one predicted by equation (1).
An earlier NI experiment (see P. W. Smith and W. J. Tomlinson, IEEE QE-20,
30 (1984)) using a quartz suspension reported a switching contrast ratio
of 1.4 through a double threshold. With a polystyrene suspension a 3 to 1
contrast was found and only a single threshold was observed. However,
agglomeration problems may have dominated the switching behavior with the
quartz suspension. This effect was significantly reduced in the
polystyrene experiment (or was at least mostly reversible) due to the
presence of surfactant and the surface charge of the polystyrene spheres.
If saturation was actually preventing the appearance of multiple
thresholds, thereby enhancing the switching quality then by modifying the
material response characteristic one should be able to manipulate the
switching behavior. For example, a higher density suspension should cause
increased switching contrast since saturation effects are more likely to
occur, resulting in reduced overshooting.
In order to test this idea an artificial Kerr medium was used having a
volume concentration of 10.5% and a similar sphere radius r=0.137 nm. This
product (obtained from Seragen Diagnostics) already contained a surfactant
concentration of 0.1-0.5% to prevent coagulation. The low intensity
refractive index of the nonlinear medium was thus offset by a calculated
value .DELTA..apprxeq.0.031 from n.sub.o and the resulting critical
glancing angle for TIR was therefore .PSI..sub.crit =12.1.degree..
Assuming the validity of equation (2) an effective nonlinear Kerr
coefficient of n.sub.2 =0.57 cm.sup.2 /MW is predicted. The switching
result obtained for this particular nonlinear medium at a glancing angle
.PSI.=7.3.degree. and theoretical focusing condition w.sub.o =4.4 nm
appears in FIG. 2. It shows the intensity dependent reflectivity curve
normalized (The denser suspension had a total reflectivity loss along the
interface on the order of 50%, while the less denser suspension had a
reduced loss of roughly 10%) versus the low intensity value, and is the
best switching curve obtained for various glancing angles. To convert from
incident power P=w.sub.o.sup.2 I.sub.max /2 normalized intensity n.sub.2
I.sub.max /.DELTA. using calculated values for n.sub.2, w.sub.o and
.DELTA., one must multiply the x-axis by a factor.apprxeq.61.
The switching contrast depended somewhat on the filling procedure for the
nonlinear medium so that FIG. has to be considered as the best observed
case and not as an absolute measurement. This dependence on the NI
preparation is another indication that the specific nonlinear response
function .DELTA.n(I) (modified by the slightly different preparations)
strongly affects the switching behavior: the normalized reflectivity
curves were typically reproducible within 5-10% over a period of 1/2 hour
if only the incident intensity was varied; they differed on the order of
10-20% if the laser intersected the interface at a different location of
the NI (uncertainty in reproducing identical focus conditions); and
differed by up to a factor 2 if various fillings, LiF crystal
repolishings, etc. were compared. Nevertheless, it was found that under
certain conditions the denser suspension produced better switching than
lower concentration levels.
Provided with this indication that saturation may well be a primary factor,
a different approach to modifying the index response function was tried:
`softening` the saturation of the induced index change. To accomplish
this, the orientation of the NI was rotated from the usual horizontal to a
vertical orientation. A redesigned NI supporting apparatus allowed
reduction in the thickness of the nonlinear medium from 2 mm to roughly
200 .mu.m and permitted a more reproducible filling procedure. These
geometry changes were intended to minimize particle deposition as well as
laser-induced particle agglomerations. The latter would tend to move out
of the laser focus simply by means of gravity, slowing the index change.
Convection due to gravity (and laser-induced heating) was maximized by
this vertical geometry further reducing the induced index change through
delocalization. Under such conditions, one would expect a higher threshold
intensity and lower switching contrast for a given maximum intensity.
The effect of rotating the NI from a horizontal to vertical orientation can
be seen in FIG. 3 which was obtained for .PSI.=7.3. with a nonlinear
liquid (Surfactant-free product of Interfacial Dynamics Corp.) [7.7 vol. %
polystyrene spheres (r=138.5 .mu.m) suspended in water with roughly 0.5%
nonionic surfactant and calculated .DELTA.=0.038, n.sub.2 =0.44 cm.sup.2
/MW, .PSI..sub.crit =13.4.degree., w.sub.o .apprxeq.5 .mu.m and
.PSI..sub.D =1.3.degree.]. FIG. 3(a) shows the best switching results
found with the horizontal NI for various glancing angles, while FIG. 3(b)
represents a typical intensity dependence obtained with the vertically
oriented NI for the same .PSI..
These changes sharply improved the reproducibility of the results. Absolute
reflectivity changed roughly 10% in the vertical orientation over a 2 hour
period while in some special cases (not FIG. 2) the horizontal
configuration showed an absolute exponential reflectivity decay with a
time constant as low as 30 minutes. Moreover, repolishing the LiF crystal,
filling of the nonlinear liquid and reoptimizing the focus typically
caused variations on the order of 10% for the vertical NI, while 10 times
larger variations were observed in the horizontal case. As expected,
threshold intensity increased and switching contrast decreased.
In summary, NI switching with enhanced contrast and a single intensity
threshold has been demonstrated. This behavior can be explained by
including saturation effects in the standard NI model. The evidence that
saturation is the primary factor responsible for the high switching
quality is not conclusive, but it seems the most likely explanation.
Further investigations, both theoretical and experimental, are necessary
for confirmation and to firmly establish the ideal shape of the nonlinear
refractive index response function. Since saturation effects in general
occur on a very short time scale, such findings will likely have a
significant impact on the application of NI's in ultrafast optical signal
processing.
The nonlinear refractive index response function, or saturation function,
can be generally designated a .DELTA.n(I). The function equals zero at
zero intensity. For positive intensity, nonlinear positive materials have
only positive values, and nonlinear negative materials have only negative
values. Further, slower nonlinear materials exhibit a reduced index change
for a given intensity. All saturation functions .DELTA.n(I) have a
limiting value called .DELTA..sub.sat for the specific saturation
function.
The first indication that improvement was possible came from introducing a
diffusion mechanism into the standard NI model (see R. Cuykendall and D.
Andersen, Opt. Comm. 62, 232 (1987) and Opt. Lett. 12, 542 (1987); R.
Cuykendall, Appl. Opt. 27, 1772 (1988); D. R. Andersen, R. Cuykendall and
J. Regan, Comp. Phys. Commun. 48, 255 (1988); R. Cuykendall and K. Strobl,
JOSAB, to be published, (1989)) delocalizing the induced index change.
Experimental findings (see K. H. Strobl, R. Cuykendall, B. Bockhop and D.
Megli, to be published in Advances in Laser Science IV, J. Gole, D. Heller
and W. C. Stwalley, editors, (American Institute of Physics, New York,
1989); K. H. Strobl and R. Cuykendall, submitted to Phys. Rev. A; K. H.
Strobl and R. Cuykendall, submitted to Opt. Lett.) strongly suggest that
saturation may also be a mechanism for achieving enhanced switching
quality. The standard NI model is herein extended to include hard-limited
saturation of the induced nonlinear refractive index change. Saturation is
shown to have a significant impact on improving the intrinsic NI switching
behavior. Since most saturation effects are virtually instantaneous, such
findings may stimulate use of the NI in novel applications.
STANDARD NI MODEL
A schematic diagram of a nonlinear interface is given in FIG. 4. The
standard (no diffusion and saturation effects) two-dimensional Gaussian
model prediction (see W. J. Tomlinson, J. P. Gordon, P. W. Smith and A. E.
Kaplan, Appl. Opt. 21, 2041 (1982); P. W. Smith and W. J. Tomlinson, IEEE
QE-20, 30 (1984); K. H. Strobl and R. Cuykendall, submitted to Phys. Rev.
A.) for an NI is based on the assumption that the light induced index
change .DELTA.n(I) is directly proportional to the local light intensity
I. The refractive index of the nonlinear medium is then given by the
formula
n=n.sub.o -.DELTA.+.DELTA.n(I)=n.sub.o -.DELTA.+n.sub.2 I (3)
where n.sub.o is the refractive index of the linear medium, .DELTA. is the
low intensity refractive index offset between the two dielectric media
comprising the NI and n.sub.2 is a Kerr-constant having the same sign as
.DELTA.. This model predicts the existence of multiple thresholds in the
intensity dependence of the NI reflectivity for a two dimensional Gaussian
beam incident on the NI from the linear medium. These thresholds are
associated with the appearance of additional transmitted (self-focused)
channels arising from additional fringes [formed by the incident and
reflected wave at the interface] reaching the threshold intensity allowing
them to switch through the interface. An example of the resulting
characteristic saw tooth reflectivity curve is shown in FIG. 5 for the
case .DELTA.=0.01, n.sub.o =1.391 and an incident Gaussian beam having an
intensity I.sub.max and a minimum 1/e amplitude radius of w.sub.o
=10.lambda.=10 .mu.m for the focus at the interface [neglecting
reflections at the interface]. The incident glancing angle was
.PSI.=7.3.degree.=0.55 .PSI..sub.crit with .PSI..sub.crit being the
critical glancing angle for plane waves. Numerical data point (small
squares) are connected with straight lines to guide the eye.
Between adjacent intensity thresholds the reflectivity increases with
intensity. The reason for this reflectivity increase is due to an
overshooting of the refractive index n.sub.o in the nonlinear medium. This
cause increased index mismatch resulting in a reduced transmission value
for the already transmitted fringes. After a couple of fringes have
already switched through the interface, the newly added fringes compensate
on the average the transmission loss of earlier transmitted fringes
resulting in a stable `average` reflectivity for intensities well above
threshold. It is unknown if this compensation is generally balanced for
any high intensity since the computer program used requires excessive
runtime [resolution] for intensities higher than the one shown in FIG. 5.
But it seems to be true between 13 and 20 times the first threshold. Note
that in contrast to these predictions of approximate stable averaged
reflectivity at high intensities for Gaussian beams, the plane wave model
(see A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977); K. H. Strobl and R.
Cuykendall, submitted to Phys. Rev. A.) shows a steady increase of the NI
reflectivity for intensity values above n.sub.2 I/.DELTA.=1.
The beam breakup connected with the multiple thresholds can be observed in
FIG. 6 which shows the calculated high resolution intensity distribution
of the parameters used to generate FIG. 5 for a normalized intensity
n.sub.2 I.sub.max /.DELTA.=11.6. The transmitted beam is broken up into 13
self-focused channels and is therefore only of limited use for optical
applications.
SATURATED NONLINEAR INTERFACE (SNI) MODEL
By extending the standard NI model to include the effect of nonlinear
refractive index saturation, the undesirable index overshooting should be
prevented so that ideally the local `self-transparensation` area at the
interface increases continuously with increasing incident intensity. The
(local) reflectivity increase would then be prevented. This can be
obtained by saturating equation (3) in such a way that transmitted fringes
can cause no or only limited overshooting of the desired index change
.DELTA.. Since in the former case the reflectivity can then only decrease,
deeper total switching should result with a pronounced initial drop.
Depending on the amount of saturation (overshooting limit), subsequent
reflectivity `oscillations` will be lessened or possibly nonexistent.
The simplest way to introduce saturation effects in the standard NI model
description is to assume the following `hard limited` Kerr-like intensity
dependency of the refractive index n for the nonlinear medium
##EQU2##
where .DELTA..sub.sat is the maximum refractive index shift which can be
induced in the nonlinear medium by a local intensity I. This very simple
saturation model allows investigation of the basic effects of saturation
on the NI switching behavior. To do this, first an investigation of how
sensitive the switching curve depends on the limiting parameter
.DELTA..sub.sat will be carried out and then a determination of the angle
and beam waist dependence of the saturation curve will follow. Unless
stated differently the parameter used throughout this paper are identical
to the one used to generate FIG. 5.
SNI dependence on the limiting value .DELTA..sub.sat. The variation of the
NI behavior when going from .DELTA..sub.sat =.DELTA. to .DELTA..sub.sat
=.infin. is shown in FIGS. 7 and 8. In the "ideal" hard limited case [FIG.
7(a)] with .DELTA..sub.sat =.DELTA. we obtain, as expected, a smooth and
steady reflectivity dropping above threshold. Slightly overshooting this
condition destroys immediately the monotony of the reflectivity curve and
causes the reappearance of additional thresholds. This behavior can be
observed when comparing FIG. 7(a) with FIG. 7(b) showing the case
.DELTA..sub.sat =1.5.DELTA.. Note the difference between the first sharp
additional threshold and the second much more diffuse one. Further
doubling of .DELTA..sub.sat to 3.DELTA. [FIG. 8(a)] increases the number
of additional thresholds from two to five and results in a notable lift of
the (average) reflectivity curve. Additional increase of .DELTA..sub.sat
produces only small changes to FIG. 8(a) so that in the case
.DELTA..sub.sat =>, displayed in FIG. 8(b), the number of additional
thresholds is only a factor of roughly 2.5 higher than in the
.DELTA..sub.sat =3.DELTA. case with nearly no additional lift of the
average reflectivity curve. Note that the most dramatic improvement in the
NI switching quality took place near the condition .DELTA..sub.sat
.apprxeq..DELTA..
The saturation influences not only the reflected beam but also the
transmitted one. Compare for example the clean, only slightly self focused
(nearly Gaussian) transmitted beam profile in FIG. 9 with the `messy` beam
break up into multiple (13) self focused `spikes` in FIG. 6. For intensity
closer to the threshold as in FIG. 9(b) the self-focusing of the
transmitted beam is more pronounced. Nevertheless the self focusing is
still significantly reduced compared to the unsaturated case.
Based on a simple plane wave picture the effect off saturating the
refractive index change should make itself notable only above the matching
point of the two refractive indices comprising the NI. This point of total
transmission (.DELTA.n(I)=.DELTA.) is at a higher intensity than the TIR
threshold. One would therefore expect the threshold to be more or less
saturation insensitive even in the Gaussian beam case, but as FIG. 7 and 8
show the SNI model predicts a more than doubling of the threshold
intensity when .DELTA..sub.sat approaches .DELTA.: the relative threshold
intensity is 2.3/1.4/1.3/1 for .DELTA..sub.sat =1/1.5/3/.infin..DELTA.. At
the first moment this behavior seems therefore counter intuitive. A more
careful analysis of this predicted saturation effect shows that the
explanation lays first in the nonplanar penetration of the evanescent wave
in the second medium comprising the NI and second in the reduction of the
gradient of the resulting local refractive index distribution through the
saturation. In the condition of TIR the incident wave penetrates
(exponentially damped) the second media, travels along the interface for a
short distance and then curves back to the incident media. This
longitudinal displacement is known as Goos Hanchen effect (see F. Goos and
H. Hanchen, Ann. Physik 1(6), 333 (1947)). Tamir (see T. Tamir, JOSA A 3,
558, 1986) has shown that for cases where the amplitude or phase change of
the reflectivity function varies most rapidly (for example near the
critical angle) a Gaussian beam reflecting at a dielectric interface can
exhibit lateral, focal, angular shifts and even a change in beam waist.
Total internal reflection at a nonlinear interface can result also in a
distortion of the reflected spatial beam profile (see A. E. Kaplan, Sov.
Phys. JETP 45 896 (1977); D. R. Andersen, R. Cuykendall and J. Regan,
Comp. Phys. Commun. 48, 255 (1988)). An investigation of the beam profiles
when approaching the TIR threshold indicates that for hard limited
Kerr-like nonlinearity the penetrated part of the incident beam can travel
for a longer distance along the interface and still bend back to the
linear medium (see FIG. 14) than in the unsaturated case. This causes the
increase in the threshold intensity retarding the switching form TIR to
partial transmission.
SNI dependence on the glancing angle .PSI.
The variation of the saturated (.DELTA..sub.sat =.DELTA.) NI switching
curve with the normalized glancing angle .PSI./.PSI..sub.crit is shown in
FIG. 10 (a)-(d) for the cases .PSI./.PSI..sub.crit =0.55, 0.60, 0.65 and
0.73. With increasing .PSI./.PSI..sub.crit a shoulder develops at the
bottom of the knee causing a small rise in reflectivity for intensity
values near the knee and a further reflectivity drop for intensities far
from it. As for the standard model, the threshold intensity drops to lower
values if .PSI./.PSI..sub.crit gets increased.
The development of a shoulder at higher glancing angle reassembles an
additional, but much less pronounced intensity threshold. This second
threshold produces quite different NI behavior than the equivalent one in
the unsaturated case. It has a staircase-like form and typically a several
times higher switching threshold. It also shows a quite different behavior
for the transmitted channels (see FIG. 11). Right after the second
threshold, instead of observing two highly self-focused channels of
roughly similar highs a big, only slightly self-focused channel and a
roughly ten times smaller one was found [FIG. 11(a)]. At even higher
intensities [FIG. 11(b)], instead of resulting in two well separated
self-focused channels, these two channels `fuse` together to form again a
more or less single channel with a little bit of structure on top of it.
Even for this less advantageous reflectivity curve the switching behavior
is definitively improved over the nonsaturated case. The development of
the additional shoulder might put some limits on the usefulness of certain
glancing angles .PSI. restricting its range for certain NI applications.
SNI dependence on the beam waist w.sub.o
A dependence of the reflectivity curve on the beam waist w.sub.o is shown
in FIG. 12. Increasing the beam waist from 5.lambda.to 8.lambda.raises the
switching contrasts [at twice the threshold intensity] roughly 34%. An
increase from w.sub.o =8.lambda.to w.sub.o =10.lambda.reduces the contrast
to roughly its w.sub.o =5.lambda.value. Further increase to w.sub.o
=15.lambda.causes a shoulder to appear resulting in an additional
reduction of the contrast by a factor of two.
The variation of the threshold intensity with the beam waist w.sub.o is
displayed in FIG. 13. The threshold varies for w.sub.o
=5/8/10/12.5/15.lambda. approximately as 5.2/1.6/1/0.67/0.55 n.sub.2
I.sub.max /.DELTA.. A least square fit to these values shows that the
threshold intensity scales (5.ltoreq.w.sub.o .ltoreq.20) roughly like
w.sub.o.sup.-2. Without saturation (.DELTA..sub.sat =.infin.) the
threshold intensity varies with the beam waist approximately as
w.sub.o.sup.-0.26. This eff | | |
