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Optical bistability near the optical Freedericksz transition
Volume 62, number
6
OPTICS
15 June 1987
COMMUNICATIONS
OPTICAL BISTABILITY NEAR THE OPTICAL FREEDERICKSZ F. MARQUIS
TRANSITION
’ and P. MEYSTRE
Optical Sciences Center, University ofArizona. Tucson, AZ 85721, USA Received
29 January
1987
We discuss dispersive optical bistability near the intrinsic second-order-type optical Freedericksz transition in a nematic liquid crystal. The combined effects of this intrinsic transition and of the first-order-type phase transition imposed by the feedback mechanism, here a Fabry-Perot resonator, leads to novel effects absent from conventional bistable systems.
Optical bistability has been observed in numerous systems, from atomic vapors to semiconductors [ I] and single electrons in traps [2]. It always relies on the combined effects of a nonlinearity and a feedback mechanism, which can be either external, typically an optical resonator, or intrinsic as is the case e.g. in optical bistability by increased absorption [ 31. In this letter, we discuss dispersive optical bistability using a medium whose intrinsic nonlinear behaviour exhibits a second-order phase transition. The combined effects of this response and of the tirstorder-type phase transition imposed by the feedback mechanism, here a Fabry-Perot resonator, leads to novel effects absent from more conventional systems. Optical bistability has previously been predicted and observed in nematic liquid crystals [4], but this work used a bias magnetic field to bring the system above the Freedericksz transition and the crystal responded essentially as a Kerr-type nonlinear medium. We consider specifically a nonlinear medium consisting of an homeotropically aligned nematic liquid crystal [ 5 ] of thickness d placed inside a Fabry-Perot cavity and irradiated by a cw laser of intensity I linearly polarized along the direction e, and propagating along the easy direction e= of the medium, perpendicular to the resonator mirrors and cell walls. We approximate the incident field by a plane wave, and ignore the transverse inhomogeneities of the
molecular reorientation. We furthermore neglect diffraction and absorption inside the crystal, and assume that the twist, splay and bend elastic constants of the crystal have the same value K. This is the so-called “one elastic constant approximation”. It was shown by Ong [ 61 that in general, nematic liquid crystals irradiated by a laser field may exhibit an intrinsic first-order phase transition, but the oneelastic constant approximation precludes the appearance of such a behaviour: in this limit, the crystal always experiences a second-order-like phase transition, the optical Freedericksz transition [ 71 when the incident intensity is increased past a threshold value It,,: For ICI,,, the average orientation of the molecules, characterized by a director angle 19(z), is along the easy axis of the crystal, e(z) = 0. But for I> Zth, the laser field induces a reorientation of the molecules in the crystal, which in turn produces a nonlinear phaseshift of the light. For small dielectric anisotropy, the director angle 19(z) is then expressed in terms of an elliptic function which for values of the field intensity close to threshold is well approximated by
’ Permanent address: Max-Planck-Institut
We restrict our discussion to this limit to avoid the complications due to the appearance of higher order
8046 Garching,
fur Quantenoptik,
6(z) =a sin(nz/d), where [7]
a=
D-
Fed. Rep. Germany
0 030-4018/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
(1)
B.V.
Zr-n21d2 3ffZrl4
’
(2)
409
Volume 62, number
OPTICS
6
modes [ 81 of the director angle field as the incident intensity is further increased. In eq. (2) Z=S_n,/c~c is the normalized z-component of the Poynting vector of the incident field, n, and n, the ordinary and extraordinary indices of refraction of the crystal, r= 1 -nzln,2 is a measure of its optical anisotropy, and (Y= 2/3 - 3r/2. Eq. (2) is valid in the limit r< 1. Nematic liquid crystals (NLC) are birefringent media with the optical axis along the main orientation B of the molecules. When a molecular reorientation is induced in the NLC, the optical axis makes therefore an angle 0(z) with the direction of propagation of light. The effective index of refraction neff seen by an extraordinary wave propagating through such a medium is [ 91
non, k(z) = [nf cos’0(z)+n~ sin28(z)]“2
’
(3)
This variation of the effective refractive index translates into a nonlinear phase shift Qn, experienced by the light as it propagates along the crystal. For the geometry at hand: d a’,,
=
F
s
(4)
0
where I is the wavelength of the incident bining eqs. (2) and (4) yields readily a,, =S(Z-I,,)/&
field. Com-
(5)
where s= [2nn,rd/3c~ll]
Tz= 1+ (R/T2)(
f
‘t---------------
/ I_/___-_
* I
11, Fig. 1. Nonlinear
phaseshift
as a function
of the input intensity.
where iDppis the linear detuning between incident laser and cavity mode frequencies and R, Tare the intensity reflection and transmission coefficients of the Fabry-Perot mirrors, with R+ T= 1 for simplicity, and we have used eq. (5). Below the optical Freedericksz transition threshold s=O and eq. (7) yields a linear relationship
10 l+(R/Tz)@;
between transmitted and incident intensities. This is the curve (L) in fig. 2. Above threshold, s is a constant, and in contrast to conventional situations in optical bistability, eq. (8) yields a quadratic, rather than a cubic relationship between Z, and Zo, specifically z = Z, + (2sRIT)( OZa+s) It,, 2 Jzi
@(Z-Z,,),
(6)
and 8 is the Heaviside function, e(x) = 0 for x< 0 and 1 for x> 0. It accounts for the absence of nonlinear effects below the Freedericksz transition, ZCZ,~. The dependence of on, on the incident intensity is illustrated in fig. 1. We now discuss the modifications to this behaviour as the system is placed inside a high finesse Fabry-Perot resonator. At steady state the intracavity intensity Z is related to the incident intensity Z. by the well-known relation Zo @Q+ @“,)2
= 1 +(RIz-2)[@,+S(z-z,,)/z]2
t
2[1+(RIT2)(@P,+s)2]
where the discriminant
’
(7)
’
lo,tll
(9)
d is
d=Z:,+(4sRIT)Zt,[Zo(@,+s)-sTI,,].
IO,,
10
410
@“I
I,=“= dz (neK-n,),
15 June 1987
COMMUNICATIONS
1”
(10)
Io,c 10.m
10
Fig. 2. Linear (L) and nonlinear (N) solutions of the resonator equation (lo), for arbitrary units. The threshold corresponds to the upper (a) or lower (b) intersection of the two curves.
Volume 62, number
OPTICS
6
15June 1987
COMMUNICATIONS
The nonlinear solutions (9) are illustrated as curve (N) in fig. 2. The condition A> 0 imposes the condition
2
-(RIT)(@,+s)]“2=Zo,,
0.6 -
(11)
on the existence of the nonlinear branch of solutions. This mathematical condition should not be confused with the threshold of the Freedericksz transition, which occurs at
(12)
Zo,,,=TI,,[l+(RI~)~~l
as readily obtained from eq. (8) and the condition that the intracavity field Z=Z,lT be larger than It,,. The solution (9) is physical only if I is larger than the OPT threshold. Thus one must distinguish the two possibilities illustrated in fig. 2. Case (a) shows the situation where the intersection between L and N corresponding to the OFT threshold ( ZO,th,I,,,) lies on the upper branch of the nonlinear characteristic. In this case the internal intensity is always a single valued function of the incident intensity, linear for I0 < Z,,,, and nonlinear for larger values of IO. In contrast, case (b) has ( ZO,th,I,,) lying on the lower branch, and multiple solutions are possible. Specifically, three solutions are possible for incident intensities in the interval ZO,cI IOI ZO.th. We emphasize that this situation is completely different from usual bistability, since only two of these solutions are related to nonlinear refraction in the optical medium. Another difference is the presence in all cases of a domain characterized by 2 solutions, for Z2Z0.th. This is related to the intrinsic threshold of the nonlinearity, which implies that the nonlinear solution has no physical sense for Z
Z(Z,,)=.SZ,,{R/[~+R(@~+.S)~]}“~. The condition
for bistability
s{R/[T~+R((~P~+s)~]}“~>
is thus satisfied 1.
(13) for
(14)
This expression yields directly the possible range of values of the linear detuning for which bistability is possible:
Fig. 3. Internal intensity as a function of the input intensity for a Fabry-Perot resonator containing a NLC. (a) @$=2. (b) op= - 10. The arrows “1” and “2” label the up- and downswitching points.
-s-(.s*-T~/R)“*I@~I
-s+(s’-T2/R)‘12. (15)
The stability analysis of the various possible solutions is straightforward, but lengthy. We simply summarize the final results here. When the input intensity is continuously increased from zero, the internal intensity increases first linearily with IO, until it reaches the OFT threshold. Past this point the linear solution becomes unstable, as would be expected intuitively. The upper branch of the nonlinear response characteristic turns out to be always stable, but (if the condition for bistability (15) is fulfilled) the part of the lower branch that satisfies ZZZ,,, corresponds to an unstable solution. These results are summarized in fig. 3, where only the physical solutions are reported. The dashed lines are unstable solutions. Case (a) is for Qp=2 and illustrates a situation where criterion (15) is not satisfied (no bistability). Case (b) on the other hand satisfies the bistability criterion, cDp= - 10. Arrow 1 corresponds to the switch-up point at Zo=Zth, where the linear solution becomes unstable. The system jumps then discontinuously to the upper nonlinear branch. When the input intensity is decreased from this point, the system stays on this branch until Z. reaches the critical value lo,= for which the switchdown to the linear solution occurs (arrow 2). It is important to emphasize once more that even 411
Volume 62, number
6
OPTICS
COMMUNICATIONS
if the resonator shows bistable regime due to the nonlinearity of the liquid crystal, the Freedericksz transition does not exhibit any intrinsic bistability in our model. The value of the internal intensity corresponding to the switch down is still higher than the threshold value of the OFT. The case where the Freedericksz transition itself shows hysteresis could be analyzed using the model of Ong, and will allow to study the effect of feedback on a system exhibiting an intrinsic first-order phase transition.
References [ I ] H.M. Gibbs, Optical bistability: [2]
[3]
[ 41 [ 51 [ 61 [ 71
Acknowledgement
The authors acknowledge financial support from the NSF-Industry Optical Circuitry Cooperative. FM also acknowledges partial financial support by the Swiss National Foundation for Scientific Research.
412
15 June 1987
[8]
[ 91
Controlling light with light (Academic Press, Orlando, 1985). A.E. Kaplan, Phys. Rev. Lett. 48 (1982) 138; G. Gabrielse, H. Dehmelt and W. Kells, Phys. Rev. Lett. 54 (1985) 537. S. Koch, H.E. Schmidt and H. Haug, J. Lumin. 30 (1985) 232. For a review of this work, see Y.R. Shen, in: Optical bistability, dynamic nonlinearity and photonic logic, eds. B.S. Wherrett and S.D. Smith (The Royal Society, London, 1985). For a review, see e.g. P.G. DeGennes, The physics of liquid crystals (Clarendon Press, Oxford, 1975). H.L. Ong, Phys. Rev. A3 I (I 985) 3450; Appl. Phys. Lett. 46 (1985) 822. B.Ya. Zeldovich, N.V. Tabiryan and Yu.S. Chilingarian, Sov. Phys. JETP 54 (1981) 32; B.Ya. Zeldovich and N.V. Tabiryan, Sov. Phys. JETP 55 (1982) 656; SD. Durbin, S.M. Arakelian and Y.R. Shen, Phys. Rev. Lett. 47 (1981) 1411. F. Marquis, P. Meystre, E.M. Wright and A.E. Kaplan, submitted to Phys. Rev. A. M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford, 1970).
6
OPTICS
15 June 1987
COMMUNICATIONS
OPTICAL BISTABILITY NEAR THE OPTICAL FREEDERICKSZ F. MARQUIS
TRANSITION
’ and P. MEYSTRE
Optical Sciences Center, University ofArizona. Tucson, AZ 85721, USA Received
29 January
1987
We discuss dispersive optical bistability near the intrinsic second-order-type optical Freedericksz transition in a nematic liquid crystal. The combined effects of this intrinsic transition and of the first-order-type phase transition imposed by the feedback mechanism, here a Fabry-Perot resonator, leads to novel effects absent from conventional bistable systems.
Optical bistability has been observed in numerous systems, from atomic vapors to semiconductors [ I] and single electrons in traps [2]. It always relies on the combined effects of a nonlinearity and a feedback mechanism, which can be either external, typically an optical resonator, or intrinsic as is the case e.g. in optical bistability by increased absorption [ 31. In this letter, we discuss dispersive optical bistability using a medium whose intrinsic nonlinear behaviour exhibits a second-order phase transition. The combined effects of this response and of the tirstorder-type phase transition imposed by the feedback mechanism, here a Fabry-Perot resonator, leads to novel effects absent from more conventional systems. Optical bistability has previously been predicted and observed in nematic liquid crystals [4], but this work used a bias magnetic field to bring the system above the Freedericksz transition and the crystal responded essentially as a Kerr-type nonlinear medium. We consider specifically a nonlinear medium consisting of an homeotropically aligned nematic liquid crystal [ 5 ] of thickness d placed inside a Fabry-Perot cavity and irradiated by a cw laser of intensity I linearly polarized along the direction e, and propagating along the easy direction e= of the medium, perpendicular to the resonator mirrors and cell walls. We approximate the incident field by a plane wave, and ignore the transverse inhomogeneities of the
molecular reorientation. We furthermore neglect diffraction and absorption inside the crystal, and assume that the twist, splay and bend elastic constants of the crystal have the same value K. This is the so-called “one elastic constant approximation”. It was shown by Ong [ 61 that in general, nematic liquid crystals irradiated by a laser field may exhibit an intrinsic first-order phase transition, but the oneelastic constant approximation precludes the appearance of such a behaviour: in this limit, the crystal always experiences a second-order-like phase transition, the optical Freedericksz transition [ 71 when the incident intensity is increased past a threshold value It,,: For ICI,,, the average orientation of the molecules, characterized by a director angle 19(z), is along the easy axis of the crystal, e(z) = 0. But for I> Zth, the laser field induces a reorientation of the molecules in the crystal, which in turn produces a nonlinear phaseshift of the light. For small dielectric anisotropy, the director angle 19(z) is then expressed in terms of an elliptic function which for values of the field intensity close to threshold is well approximated by
’ Permanent address: Max-Planck-Institut
We restrict our discussion to this limit to avoid the complications due to the appearance of higher order
8046 Garching,
fur Quantenoptik,
6(z) =a sin(nz/d), where [7]
a=
D-
Fed. Rep. Germany
0 030-4018/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
(1)
B.V.
Zr-n21d2 3ffZrl4
’
(2)
409
Volume 62, number
OPTICS
6
modes [ 81 of the director angle field as the incident intensity is further increased. In eq. (2) Z=S_n,/c~c is the normalized z-component of the Poynting vector of the incident field, n, and n, the ordinary and extraordinary indices of refraction of the crystal, r= 1 -nzln,2 is a measure of its optical anisotropy, and (Y= 2/3 - 3r/2. Eq. (2) is valid in the limit r< 1. Nematic liquid crystals (NLC) are birefringent media with the optical axis along the main orientation B of the molecules. When a molecular reorientation is induced in the NLC, the optical axis makes therefore an angle 0(z) with the direction of propagation of light. The effective index of refraction neff seen by an extraordinary wave propagating through such a medium is [ 91
non, k(z) = [nf cos’0(z)+n~ sin28(z)]“2
’
(3)
This variation of the effective refractive index translates into a nonlinear phase shift Qn, experienced by the light as it propagates along the crystal. For the geometry at hand: d a’,,
=
F
s
(4)
0
where I is the wavelength of the incident bining eqs. (2) and (4) yields readily a,, =S(Z-I,,)/&
field. Com-
(5)
where s= [2nn,rd/3c~ll]
Tz= 1+ (R/T2)(
f
‘t---------------
/ I_/___-_
* I
11, Fig. 1. Nonlinear
phaseshift
as a function
of the input intensity.
where iDppis the linear detuning between incident laser and cavity mode frequencies and R, Tare the intensity reflection and transmission coefficients of the Fabry-Perot mirrors, with R+ T= 1 for simplicity, and we have used eq. (5). Below the optical Freedericksz transition threshold s=O and eq. (7) yields a linear relationship
10 l+(R/Tz)@;
between transmitted and incident intensities. This is the curve (L) in fig. 2. Above threshold, s is a constant, and in contrast to conventional situations in optical bistability, eq. (8) yields a quadratic, rather than a cubic relationship between Z, and Zo, specifically z = Z, + (2sRIT)( OZa+s) It,, 2 Jzi
@(Z-Z,,),
(6)
and 8 is the Heaviside function, e(x) = 0 for x< 0 and 1 for x> 0. It accounts for the absence of nonlinear effects below the Freedericksz transition, ZCZ,~. The dependence of on, on the incident intensity is illustrated in fig. 1. We now discuss the modifications to this behaviour as the system is placed inside a high finesse Fabry-Perot resonator. At steady state the intracavity intensity Z is related to the incident intensity Z. by the well-known relation Zo @Q+ @“,)2
= 1 +(RIz-2)[@,+S(z-z,,)/z]2
t
2[1+(RIT2)(@P,+s)2]
where the discriminant
’
(7)
’
lo,tll
(9)
d is
d=Z:,+(4sRIT)Zt,[Zo(@,+s)-sTI,,].
IO,,
10
410
@“I
I,=“= dz (neK-n,),
15 June 1987
COMMUNICATIONS
1”
(10)
Io,c 10.m
10
Fig. 2. Linear (L) and nonlinear (N) solutions of the resonator equation (lo), for arbitrary units. The threshold corresponds to the upper (a) or lower (b) intersection of the two curves.
Volume 62, number
OPTICS
6
15June 1987
COMMUNICATIONS
The nonlinear solutions (9) are illustrated as curve (N) in fig. 2. The condition A> 0 imposes the condition
2
-(RIT)(@,+s)]“2=Zo,,
0.6 -
(11)
on the existence of the nonlinear branch of solutions. This mathematical condition should not be confused with the threshold of the Freedericksz transition, which occurs at
(12)
Zo,,,=TI,,[l+(RI~)~~l
as readily obtained from eq. (8) and the condition that the intracavity field Z=Z,lT be larger than It,,. The solution (9) is physical only if I is larger than the OPT threshold. Thus one must distinguish the two possibilities illustrated in fig. 2. Case (a) shows the situation where the intersection between L and N corresponding to the OFT threshold ( ZO,th,I,,,) lies on the upper branch of the nonlinear characteristic. In this case the internal intensity is always a single valued function of the incident intensity, linear for I0 < Z,,,, and nonlinear for larger values of IO. In contrast, case (b) has ( ZO,th,I,,) lying on the lower branch, and multiple solutions are possible. Specifically, three solutions are possible for incident intensities in the interval ZO,cI IOI ZO.th. We emphasize that this situation is completely different from usual bistability, since only two of these solutions are related to nonlinear refraction in the optical medium. Another difference is the presence in all cases of a domain characterized by 2 solutions, for Z2Z0.th. This is related to the intrinsic threshold of the nonlinearity, which implies that the nonlinear solution has no physical sense for Z
Z(Z,,)=.SZ,,{R/[~+R(@~+.S)~]}“~. The condition
for bistability
s{R/[T~+R((~P~+s)~]}“~>
is thus satisfied 1.
(13) for
(14)
This expression yields directly the possible range of values of the linear detuning for which bistability is possible:
Fig. 3. Internal intensity as a function of the input intensity for a Fabry-Perot resonator containing a NLC. (a) @$=2. (b) op= - 10. The arrows “1” and “2” label the up- and downswitching points.
-s-(.s*-T~/R)“*I@~I
-s+(s’-T2/R)‘12. (15)
The stability analysis of the various possible solutions is straightforward, but lengthy. We simply summarize the final results here. When the input intensity is continuously increased from zero, the internal intensity increases first linearily with IO, until it reaches the OFT threshold. Past this point the linear solution becomes unstable, as would be expected intuitively. The upper branch of the nonlinear response characteristic turns out to be always stable, but (if the condition for bistability (15) is fulfilled) the part of the lower branch that satisfies ZZZ,,, corresponds to an unstable solution. These results are summarized in fig. 3, where only the physical solutions are reported. The dashed lines are unstable solutions. Case (a) is for Qp=2 and illustrates a situation where criterion (15) is not satisfied (no bistability). Case (b) on the other hand satisfies the bistability criterion, cDp= - 10. Arrow 1 corresponds to the switch-up point at Zo=Zth, where the linear solution becomes unstable. The system jumps then discontinuously to the upper nonlinear branch. When the input intensity is decreased from this point, the system stays on this branch until Z. reaches the critical value lo,= for which the switchdown to the linear solution occurs (arrow 2). It is important to emphasize once more that even 411
Volume 62, number
6
OPTICS
COMMUNICATIONS
if the resonator shows bistable regime due to the nonlinearity of the liquid crystal, the Freedericksz transition does not exhibit any intrinsic bistability in our model. The value of the internal intensity corresponding to the switch down is still higher than the threshold value of the OFT. The case where the Freedericksz transition itself shows hysteresis could be analyzed using the model of Ong, and will allow to study the effect of feedback on a system exhibiting an intrinsic first-order phase transition.
References [ I ] H.M. Gibbs, Optical bistability: [2]
[3]
[ 41 [ 51 [ 61 [ 71
Acknowledgement
The authors acknowledge financial support from the NSF-Industry Optical Circuitry Cooperative. FM also acknowledges partial financial support by the Swiss National Foundation for Scientific Research.
412
15 June 1987
[8]
[ 91
Controlling light with light (Academic Press, Orlando, 1985). A.E. Kaplan, Phys. Rev. Lett. 48 (1982) 138; G. Gabrielse, H. Dehmelt and W. Kells, Phys. Rev. Lett. 54 (1985) 537. S. Koch, H.E. Schmidt and H. Haug, J. Lumin. 30 (1985) 232. For a review of this work, see Y.R. Shen, in: Optical bistability, dynamic nonlinearity and photonic logic, eds. B.S. Wherrett and S.D. Smith (The Royal Society, London, 1985). For a review, see e.g. P.G. DeGennes, The physics of liquid crystals (Clarendon Press, Oxford, 1975). H.L. Ong, Phys. Rev. A3 I (I 985) 3450; Appl. Phys. Lett. 46 (1985) 822. B.Ya. Zeldovich, N.V. Tabiryan and Yu.S. Chilingarian, Sov. Phys. JETP 54 (1981) 32; B.Ya. Zeldovich and N.V. Tabiryan, Sov. Phys. JETP 55 (1982) 656; SD. Durbin, S.M. Arakelian and Y.R. Shen, Phys. Rev. Lett. 47 (1981) 1411. F. Marquis, P. Meystre, E.M. Wright and A.E. Kaplan, submitted to Phys. Rev. A. M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford, 1970).