1. INTRODUCTIONOrganic media are important subjects of study for use inphotonics devices. In recent years much research on theoptical properties of organic media has been conducted,1
with the common understanding that organic media havethe advantages over semiconductors and inorganic mediaof faster optical response time, diversified constructionand formation, and higher optical damage thresholds.The phenomenon of optical bistability in organic mediahas interesting prospects for applications in optoelectron-ics and optical computers. We know that optical bista-bilities in semiconductors have been studied widely forseveral years.2–8 With respect to bistability in organicmedia, a theory of optical bistability in a band model ofdye molecules was presented,9,10 and bistable behavior insome organic media was experimentally displayed.11–14
In 1993 optical bistability in fullerenes was reported.15
A nonlinear Fabry–Perot (FP) cavity filled with a thinpolystyrene layer doped with C60 /C70 showed a bistableeffect at a wavelength of 1.06 mm. Fullerenes can betreated as four-level systems with both single states andtriplet states. Their resonant absorption wavelengths ofsingle states and triplet states are far different from thelaser wavelength of 1.06 mm. As far as we know, notheory has been proposed to explain such bistability withlarge dispersion. It is necessary to propose a theoreticalmodel of optical bistability including both absorptive anddispersive mechanisms in the four-level system. Such afour-level model with single and triplet states is typical of
organic media, is important for our understanding of theeffect of triplet states on bistable behavior with a nonreso-nant interacting laser, and is helpful when one selects theoperating laser wavelength and the kinds of organic me-dia with different absorption characteristics and for de-signs of the optical cavity.It is known that theories of optical bistability through
absorption and dispersion in a two-level system have beenwidely studied.16–21 By means of the rate-equation ap-proximation of a density matrix, we have studied the ef-fect of triplet-state absorption on optical bistability insuch a four-level system.22 For the rate-equation ap-proximation we assumed that the system had a smalltransition detuning. Then the results presented in Ref.22 were valid with the limitation of small transition de-tuning. We cannot explain the dispersive bistable behav-ior with large transition detuning of single and tripletstates. For more comprehensive results without the ap-proximation of small transition detuning we should treatthis problem with a semiclassical equation.In this paper we derive the susceptibility of this four-
level system with single and triplet states by solving theequation of the density matrix. In addition, with thetheory of an interferometer, we obtain the equation of op-tical bistability. With this bistability equation we canthen study optical bistability numerically under the con-ditions of dispersion and absorption of single and tripletstates. We also discuss the effects of the cooperative pa-rameters, resonant transition detuning parameters, and
1997 Optical Society of America
1110 J. Opt. Soc. Am. B/Vol. 14, No. 5 /May 1997 Ji et al.
cavity detuning parameters on optical-bistability behav-ior. This theoretical model can explain not only purelyabsorptive bistability but also purely dispersive bistabil-ity. It is more comprehensive and exact than the modelproposed in Ref. 22. Some results are different, even op-posite from those reported in Ref. 22.
2. THEORY OF OPTICAL BISTABILITYThe four-level model is depicted in Fig. 1. Levels 1 and 2are single ground and excited states, respectively. Lev-els 3 and 4 are triplet states. m21 is the electric dipolemoment between state 1 and state 2. m43 is that betweenstate 3 and state 4. g21 is the relaxation rate from level 2to level 1, and g43 is that from level 4 to level 3. g is thetransition rate of an intersystem crossing from singlestate 2 to triplet state 3. Even the electric dipole transi-tion between levels 1 and 3 is forbidden; there is collisionrelaxation from level 3 to level 1 with probability T21.We consider a linearly polarized light field with wave
number k0 and frequency v interacting with the closedfour-level system. The electric field intensity is writtenas E 5 (E0/2)@exp(ivt) 1 exp(2ivt)#. Then we have V215 (2m21E0/2)@exp(ivt) 1 exp(2ivt)# and V43 5 (2m433 E0/2)@exp(ivt) 1 exp(2ivt)#. V21 is the interactive po-tential of the light field with the electric dipole momentbetween levels 1 and 2. V43 is that between levels 3 and4. With the rotating-wave approximation and the slowlyvarying amplitude approximation the density-matrixequation related to the macroscopic electric polarizationcan be written as
r11 5 r33 /T 1 r22g21 2m21E0i2\
~r12 2 r21!,
r22 5 2r22g 2 r22g21 1m21E0i2\
~r12 2 r21!,
r33 5 2r33 /T 1 r22g 1 g43r44 2m43E0i2\
~r34 2 r43!,
r44 5 2g43r44 1m43E0i2\
~r34 2 r43!,
Fig. 1. Energy-level diagram. m21 is the dipole matrix elementbetween levels 1 and 2, and m43 is that between levels 3 and 4. gis the transition rate of the intersystem crossing. g43 is the re-laxation rate from level 4 to level 3, and g21 is that from level 2 tolevel 1. T21 is the collapse relaxation probability from level 3 tolevel 1.
r21 5 2@i~v21 2 v! 1 g'#r21 2m21E0i2\
~r22 2 r11!,
r43 5 2@i~v43 2 v! 1 g'8#r43 2m43E0i2\
~r44 2 r33!,
r11 1 r22 1 r33 1 r44 5 1, (1)
along with the equations of their complex conjugates.v21 is the resonant transition frequency between levels 1and 2. v43 is that between levels 3 and 4. r21 and r43are slowly varying amplitudes of r21 and r43 , respectively.g' 5 1/2(g 1 g21) 1 gph is the decay rate of r21 , and g'85 1/2@g43 1 (1/T)# 1 gph8 is that of r43 . gph and gph8 aredephasing rates. The positive frequency part of the slow-ing varying polarization is expressed by
P ~1! 5 N~m43r43 1 m21r21! 5 ~E0/2!e0x, (2)
where e0 is the dielectric constant, x is the complex dielec-tric susceptibility, and N is the particle number density ofa four-level system.In solving the time-dependent Eqs. (1) we handle the
problem dynamically. It is a significant subject of bista-bility with a pulsed laser, but there is more tedious alge-bra and no simple analytic expression for the parametersof interest. Here we simply treat this problem as steadystate under specific conditions. First, we assume that r44is approximately zero compared with r33 , which meansthat the generated occupancy of upper state 4 is smallcompared with that of lower state 3 at the E0 values ofinterest. Second, we assume that g @ 2/T is satisfied,which means that the triple states are long lived. Thenwe can get a relatively simple expression of the electricpolarization by solving Eqs. (1) and (2) in the steady state:
P ~1! 5iE0e0a0~1 1 id12!
2k0~1 1 d122 1 E0
2/Es2!
1iE0e0aT~1 1 id34!E0
2/Es2
2k0~1 1 d342!~1 1 d12
2 1 E02/Es
2!, (3)
where a0 5 Nm212k0 /(\e0g') is the resonant absorption
coefficient from level 1 to level 2 and aT5 Nm43
2k0 /(\e0g'8) is that from level 3 to level 4. Thedetuning parameters between the light field and the reso-nant transition are d12 5 (v 2 v21)/g' and d34 5 (v2 v43)/g'8, respectively. The saturation light field isgiven by Es
2 5 2\2g'(g 1 g21)/(m212gT). It is caused
only by the transition associated with ground-state ab-sorption. Then the complex dielectric susceptibility is
x 5 2F a0d12
k0~1 1 d122 1 E0
2/Es2!
1aTd34E0
2/Es2
k0~1 1 d342!~1 1 d12
2 1 E02/Es
2!G
1 iF a0
k0~1 1 d122 1 E0
2/Es2!
1aTE0
2/Es2
k0~1 1 d342!~1 1 d12
2 1 E02/Es
2!G . (4)
Ji et al. Vol. 14, No. 5 /May 1997/J. Opt. Soc. Am. B 1111
We consider a FP cavity filled with the four-level-system medium. The FP cavity length is L, and the re-flection coefficients of the mirrors are R. In the case oflow loss and a high-Q cavity, taking the mean-field ap-proximation, we get the following relationship betweenincident field Ei and transmitted field Et : Et 5 @(12 R)Ei]/@1 2 R exp(i2nk0L)#, where n 5 1 1 x/2 is thecomplex refractive index. Expanding the exponentialterm of this equation to the first-order term and neglect-ing the multiplication of the two first-order terms, we get
Et 5~1 2 R !Ei
1 2 R~1 1 ik0Lx 2 iu!, (5)
where the cavity detuning is given by u 5 2Mp2 2k0L. M is an integer. Introducing Eq. (4) into Eq.(5), we find that the optical bistability equation in thefour-level system is
y2 5 x2H F1 12C0
1 1 d122 1 x2
12CTx
2
~1 1 d342!~1 1 d12
2 1 x2!G2
1 FU 12C0d12
1 1 d122 1 x2
12CTd34x
2
~1 1 d342!~1 1 d12
2 1 x2!G2J , (6)
where x2 5 Et2/@(1 2 R)Es
2# is the dimensionless fieldamplitude related to the transmitted intensity and y2
5 Ei2/@(1 2 R)Es
2# is that related to the incident inten-sity. If R 5 0.96, the incident field Ei is 0.5 to 10 timesof the saturation field Es and y is 2.5–50. The relation-ship between E0 and Et is E0 5 Et /A(1 2 R). The coop-erative parameters are defined as 2C0 5 a0LR/(1 2 R)and 2CT 5 aTLR/(1 2 R) and are proportional to theabsorption coefficients of single and triplet states, respec-tively. The cavity detuning parameter is defined as U5 uR/(1 2 R). If we set R 5 0.99 for a high-Q cavityand set u 5 60.1, which is small compared to unity, asdefined above, U will be 29.9–19.9.
3. PURELY DISPERSIVE OPTICALBISTABILITYEquation (4) describes the purely dispersive dielectricsusceptibility when the absorption term of the imaginarypart is approximately zero compared with the dispersionterm of the real part. When ud12u @ 1 and ud34u @ 1, thedetuning of the transition of single and triplet states is solarge that the absorption of single and triple states can beneglected. We have
d34 /~1 1 d342! ' 1/d34 . (7)
The refractive index of pure dispersion from Eq. (4) isgiven by
n 5 1 2a0d12
2k0~1 1 d122 1 E0
2/Es2!
2aTE0
2/Es2
2k0d34~1 1 d122 1 E0
2/Es2!. (8)
From Eq. (6) we get the bistability equation of pure dis-persion:
y2 5 x2H 1 1 FU 12C0d12
1 1 d122 1 x2
12CTx
2
d34~1 1 d122 1 x2!
G2J . (9)
The bistability curves are shown in Fig. 2. The solidcurves indicate the bistability of the four-level system.Their parameters are C0 5 65, d12 5 25, d34 5 250, U5 0, and CT 5 70, 40, 20. The purely dispersive bista-bility of a two-level system is shown by the dashed curvewith CT 5 0. With the increase of CT from 0 to 70 thehysteresis loops increase. Comparing the solid curveswith the dashed curve, we find that the solid curves of thefour-level system have larger hysteresis loops than thedashed curve of the two-level system. The bistable be-havior of the four-level system is more obvious than thatof the two-level system. In addition, the four-level sys-tem of solid curves needs little incident light to switch onbistability, whereas the two-level system of the dashedcurve needs more incident light intensity to switch on bi-stability. Equation (8) indicates that the refractive indexof the four-level system has an additional positive term(aT Þ 0, d34 , 0) than the refractive index of the two-level system (aT 5 0). So the bistabilities of the four-level system display a special behavior different from thatof the two-level system. In the solid curve with CT5 70, CT . C0 or aT . a0 is satisfied. We come to theconclusion that the reverse saturable absorptive mediumcan display purely dispersive optical bistable behavior.
Fig. 2. Purely dispersive bistability curves. Solid curves arefor C0 5 65, d12 5 25, d34 5 250, U 5 0, and the values of CTshown. The dashed curve with CT 5 0 and d34 5 0 representsthe case of a two-level system.
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Figure 3 shows the effect of C0 on bistable behavior inpure dispersion. Constants are CT 5 40, d12 5 25, d345 250, and U 5 0. The values of C0 are 40, 50, 65, and75. The dashed curve with C0 5 65 is the same as thecurve with CT 5 40 in Fig. 2. With the increase of C0the hysteresis loops increase. So does the incident-lightintensity for switching on bistability. In Figs. 2 and 3both CT and C0 are beneficial to the hysteresis loops, butthey have an opposite effect on the threshold of incident-light intensity for switching on bistability.In the case of ud34u @ 1 and ud12u2 @ 1 1 E0
2/Es2 we
have
d12
1 1 d122 1 E0
2/Es2 '
1
d122
E02/Es
2
d123 , (10)
E02/Es
2
1 1 d122 1 E0
2/Es2 '
1 1 E02/Es
2
d124 1
E02/Es
2
d122 .
(11)
From Eq. (4) the refractive index is given by
n 5 1 2a0
2k0d122
aT
2k0d124d34
1E0
2
Es2 S a0
2k0d123
2aT
2k0d122d34
2aT
2k0d124d34
D . (12)
It is in the form of a Kerr medium. So the bistabilityequation of this Kerr medium is
y2 5 x2F1 1 S U 12C0
d121
2CT
d34d124 2
2C0x2
d123 1
2CT x2
d34d122
12CT x
2
d34d124D 2G . (13)
Equation (13) describes the purely dispersive bistable be-havior in the case of very low intensity of the light field(1 1 E0
2/Es2 ! ud12u2).
Fig. 3. Purely dispersive bistability curves. Curves are forCT 5 40, d12 5 25, d34 5 250, U 5 0, and values of C0 shown.
4. OPTICAL BISTABILITY WITHDISPERSION OF TRIPLE STATESWith d34 Þ 0 and d12 5 U 5 0, from Eq. (6) the bistabil-ity equation is
y2 5 x2H F1 12C0
1 1 x21
2CT x2
~1 1 d342!~1 1 x2!
G21 F 2CTd34x
2
~1 1 d342!~1 1 x2!
G2J . (14)
The effect of d34 on bistable behavior is shown in Fig. 4.Different values of ud34u are 0, 1, and 3. Constants areC0 5 10, CT 5 0.2, and d12 5 U 5 0. The dashed curvewith ud34u 5 0 represents the purely absorptive bistabil-ity. With the increase of ud34u from 0 to 3 the hysteresisloops increase. This bistable behavior is completely dif-ferent from that analyzed before.22 In the case of d34@ 1 the absorption of triplet states is approximately zerocompared with the dispersion of triple states. From Eq.(6) and approximation (7) the bistability equation is
Fig. 4. Bistability curves for various values of ud34u, with C05 10, CT 5 0.2, and d12 5 U 5 0.
Fig. 5. Bistability curves for various values of ud12u, with C05 10, CT 5 0.2, and d34 5 U 5 0.
Ji et al. Vol. 14, No. 5 /May 1997/J. Opt. Soc. Am. B 1113
y2 5 x2H F1 12C0
1 1 x2G2
1 F 2CT x2
d34~1 1 x2!G2J . (15)
In Fig. 4 the curve with ud34u 5 15 depicts this case. Itrepresents the tendency of the bistability curve withud34u @ 1.
5. OPTICAL BISTABILITY WITHDISPERSION OF SINGLE STATESWith d12 Þ 0 and d34 5 U 5 0, Eq. (6) becomes
y2 5 x2F S 1 12C0 1 2CT x
2
1 1 d122 1 x2 D
2
1 S 2C0d12
1 1 d122 1 x2D
2G .(16)
The effect of d12 on bistability is shown in Fig. 5. Thecurves are for ud12u 5 0, 1, 2, with constants C0 5 10,CT 5 0.2, and d34 5 U 5 0. The dashed curve withud12u 5 0 is the same as the dashed curve with d34 5 0 inFig. 4. With the increase of ud12u the hysteresis loops de-crease. This bistable behavior is completely differentfrom that analyzed before.22 In the case of ud12u @ 1 theabsorption of single states is approximately zero com-pared with the dispersion of single states. From Eq. (6)the optical-bistability equation can be written as
y2 5 x2F S 1 12CT x
2
1 1 d122 1 x2D
2
1 S 2C0d122
1 1 d122 1 x2D
2G .(17)
With ud12u2 @ 1 1 x2, the four-level medium is a Kerr me-dium. From Eq. (6) and approximations (10) and (11) wehave the optical-bistability equation:
y2 5 x2H F1 1 2CTS 1 1 x2
d124 1
x2
d122D G2
1 F2C0S 1
d122
x2
d123D G2J . (18)
6. DISPERSIVE AND ABSORPTIVEOPTICAL BISTABILITYThe bistabilities with both dispersion and absorption inthe four-level system display complex behavior. For clar-ity and convenience we take a typical example with C05 10, CT 5 0.2, d12 5 21, d34 5 1, and U 5 0 as shownby dashed curves. According to Eq. (6) we study numeri-cally the effects of d12 , d34 , and U on bistable behavior oftypical example, as follows.In Fig. 6(a) d12 is both less and more than the typical
value of 21, its values being 22.5, 0, 1, and 2.5. Thedashed curve describes the typical example with d125 21. In Fig. 6(b) d12 varies continually from 24 to 4.We find that the hysteresis loops increase initially andthen decrease when d12 is increased from 22.5 to 2.5. Asin Fig. 5 with d34 5 0, the hysteresis loops decrease withan increase of ud12u. The curve with d12 5 0 has the larg-est hysteresis loop. However, Fig. 6(a) indicates clearlythat the positive and negative values of d12 , with equalabsolute value, affect bistable behavior differently. Webelieve that this effect is caused by the dispersion of triple
states. d12 has a fixed range for the realization of bista-bility. Beyond this range the bistability turns into singlestability. In Fig. 6(b), which is a surface plot of the bi-stability curves, this range is in the area of the convexcamber near the zero of d12 . The conclusion obtainedhere is opposite that for d12 in Ref. 22.The effect of d34 on bistability is shown in Fig. 7. In
Fig. 7(a), d34 is chosen to be 23, 21, 0, and 3, both lessand more than the typical value of 1. The dashed curvewith d34 5 1 describes the typical example, as does thedashed curve with d12 5 21 in Fig. 6(a). In Fig. 7(b), d34varies from 24 to 4. We find that the hysteresis loopsdecrease initially and then increase with an increase ofd34 from 23 to 3. The curve with d34 5 0 has the small-est hysteresis loop. As in Fig. 4 with d12 5 0, the hyster-esis loops increase with an increase of ud34u. When ud34uincreases further, the bistability curves tend to the limitcurve. The hysteresis loops do not increase indefinitely.
Fig. 6. (a) Bistability curves for various values of d12 , with C05 10, CT 5 0.2, d34 5 1, and U 5 0. (b) Bistability curves forcontinuous values of d12 from 24 to 4, with C0 5 10, CT 5 0.2,d34 5 1, and U 5 0.
1114 J. Opt. Soc. Am. B/Vol. 14, No. 5 /May 1997 Ji et al.
Figure 7(b) shows that beyond the concave grooved areaof the surface plot around the zero of d34 the two wings ofthe surface plot beside the concave, grooved area extendmore and more evenly as d34 increases in positive andnegative directions. Unlike for Fig. 4, we know from Fig.7(a) that the positive and negative values of d34 , withequal absolute value, do not affect bistable behavior to thesame degree. This effect is caused by the dispersion ofsingle states.The effect of U on bistability is shown in Fig. 8. In Fig.
8(a), U is both less and more than the typical value of 0, at23, 22, 21, 1, 2, and 3. The dashed curve with U 5 0describes the same typical example as the two dashedcurves in Figs. 6(a) and 7(a). In Fig. 8(b), U varies from24 to 4. We find that the hysteresis loops decrease withan increase of uUu. The smaller uUu is, the larger the hys-teresis loop is. The curve with U 5 0 has the largesthysteresis loop. The value of U obviously affects the
Fig. 7. (a) Bistability curves for various values of d34 , with C05 10, CT 5 0.2, d12 5 21, and U 5 0. (b) Bistability curvesfor continuous values of d34 from 24 to 4, with C0 5 10, CT5 0.2, d12 5 21, and U 5 0.
threshold of incident-light intensity for switching on bi-stability.Through the discussion above, we find that for the
physical parameters of d12 , d34 , and U the positive andnegative values, with equal absolute value, have differenteffects on bistability. The specific characteristic of non-symmetry is a result of the interaction of the single-statestransition, triple-states transition, and a cavity with laserlight.
7. PURELY ABSORPTIVE OPTICALBISTABILITYWith d12 5 d34 5 0 and U 5 0, Eq. (6) reduces to
y 5 xS 1 12C0 1 2CT x
2
1 1 x2 D . (19)
Fig. 8. (a) Bistability curves for various values of U, with C05 10, CT 5 0.2, d12 5 21, and d34 5 1. (b) Bistability curvesfor continuous values of U from 24 to 4, with C0 5 10, CT5 0.2, d12 5 21, and d34 5 1.
Ji et al. Vol. 14, No. 5 /May 1997/J. Opt. Soc. Am. B 1115
The requirement for the existence of the bistable behavioris given by22 C0 . 9CT 1 4. The effect of CT on bistablebehavior is shown in Fig. 9, where C0 5 10 is constantand CT is chosen to be 0, 0.2, 0.4, and 0.7. The dashedcurve with CT 5 0 represents bistability in a two-levelsystem. With the increase of CT from 0 to 0.7, the hys-teresis loops decrease. With CT 5 0.7 the bistability dis-appears, and with CT 5 0 the hysteresis loop is the larg-est. The physical explanation of purely absorptivebistability is given in the Ref. 22. We can say that CT isdetrimental to bistability in purely absorptive bistability.However, in the case of purely dispersive bistabilityshown in Fig. 2, the larger CT is, the bigger the hysteresisloop. CT is beneficial to bistability in pure dispersion.CT or aT [2T 5 2CT(1 2 R)/(LR)], the resonant absorp-tion coefficient of triple states, has contrary action on dis-persive bistability and absorptive bistability.With cavity detuning U Þ 0, Eq. (19) becomes
y2 5 x2F S 1 12C0 1 2CT x
2
1 1 x2 D 2 1 U2G . (20)
Fig. 9. Bistability curves for various values of CT , with C05 10 and d12 5 d34 5 U 5 0. The dashed curve corresponds tothe case of the two-level system.
Fig. 10. Bistability curves for various values of uUu, with C05 10, CT 5 0.2, and d12 5 d34 5 0.
The requirement for the existence of bistable behavior isgiven by22
~2C0 1 1 !2~C0 2 9CT 2 4 !
27~C0 2 CT!. U2 . 0, (21)
where C0 . 9CT 1 4 is also satisfied. In Fig. 10 differ-ent values of uUu are 0, 1, and 2, and constants are C05 10, CT 5 0.2, and d12 5 d34 5 0. The dashed curvewith U 5 0 is the same as the dashed curves in Figs. 4and 5. We find that with the increase of uUu, the hyster-esis loops decrease. The dashed curve with uUu 5 0 hasthe largest hysteresis loop. The positive and negativevalues of U, with equal absolute value, have the same ef-fect on bistable behavior. This effect is different fromthat in Fig. 8 with d12 5 21 and d34 5 1.
8. CONCLUSIONWe have introduced a theoretical model of bistability in afour-level system with single and triplet states typical oforganic media. Both dispersive and absorptive mecha-nisms are included. Then our comprehension of bistabil-ity in such a four-level system is complete. With analysisof the equation of bistability we can obtain the variouskinds of mathematical description of bistability in thefour-level system. The condition that the bistabilityequation has two positive, real, extreme values of y canalso be obtained. This requirement defines the thresholdfor the existence of bistable behavior and formulates therelationship among the physical parameters C0 , CT , d12 ,d34 , and U. The relationship functions of these five pa-rameters are complex and not analytic. However, a nu-merical method is available. We do not discuss the prob-lem further in this paper.The physical comprehension of this bistable behavior is
based on an understanding of absorption and dispersionand associated detuning of such a four-level system (andsome understanding of pairs of a two-level system) and onan understanding of FP optical bistability. Because thecooperative parameters of C are proportional to the ab-sorption coefficient a and the length of the FP cavity L asdefined above, we can discuss the effect of a0 , aT , and Lon bistable behavior. d12 and d34 can be explained as thefrequency difference between the frequency of light andthe frequency of a resonant transition, normalized withthe width of line-shape function of absorption. The gen-eral bistability can be considered the composition of dis-persive bistability and absorptive bistability. Our con-clusions are as follows:
1. The absorption between triplet states is beneficialto dispersive bistability but is detrimental to absorptivebistability, as is shown by enlargement or decrease of thehysteresis loop of the bistability curve. The absorptionbetween single states is beneficial to both dispersive andabsorptive bistability.2. In dispersive bistability, an increase of the absorp-
tion between single states increases the threshold ofincident-light intensity for switching on bistability,whereas an increase of the absorption between tripletstates decreases the threshold. In absorptive bistability,
1116 J. Opt. Soc. Am. B/Vol. 14, No. 5 /May 1997 Ji et al.
the absorption between both single states and tripletstates is detrimental to the decrease of the threshold.3. The length of FP cavity is beneficial to the increase
of hysteresis loops in dispersive bistability but is detri-mental to the decrease of the threshold for switching onbistability in absorptive bistability. Although the lengthof the cavity has no relation to the refractive index of themedium, it does affect the bistable behavior. One shouldconsider this conclusion in designing the FP cavity for bi-stability.4. With an increase of ud12u, the hysteresis loops de-
crease. In the case of d12 5 0, i.e., when the frequency oflight is equal to that of the resonant transition betweensingle states, the hysteresis loop is the largest. However,with an increase of ud34u, the hysteresis loops increase.The larger the deviation of the frequency of light fromthat of the resonant transition between triplet states, thelarger the hysteresis loop; it tends to be an extreme value.5. With an increase of uUu, the hysteresis loops de-
crease. U has an obvious effect on the threshold ofincident-light intensity for switching on bistability. Thechange of resonant frequency of the FP cavity owing tothe undulation of the cavity length will modify thebistable behavior. So will a shift of the laser frequency.6. The physical parameters of d12 , d34 , and U have a
nonsymmetry effect on bistable behavior; that is, the posi-tive and the negative values, with equal absolute value,affect bistability differently.
Figure 11 shows the effect of parameters on hysteresis
Fig. 11. Effects of parameters on the hysteresis loop of a bista-bility curve. With an increase of relative parameters along thedirections of the arrows, the hysteresis loop increases.
Fig. 12. Effect of parameters on the threshold of incident-lightintensity for switching on bistability. With an increase of rela-tive parameters along the directions of the arrows, the thresholddecreases.
loops of bistability. The curves about the single staterepresent C0g(v), and those about the triplet state repre-sent CTG(v). g(v) is the line-shape function of the ab-sorption between single states. G(v) is that betweentriplet states. With the increase of relative parametersalong the directions of the arrows, the hysteresis loop in-creases. Figure 12 shows the effects of C0 and CT on thethreshold of incident-light intensity for switching on bi-stability. With the increase of relative parameters alongthe directions of the arrows, the threshold decreases.The four-level structure with single states and long-
lived triplet states proposed here is typical of organic me-dia. The theory of bistability of this system should behelpful for experimental investigation and for the devel-opment of novel, high-quality bistability devices of or-ganic media.
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