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Transverse effects in absorptive optical bistability
Volume 44, number 4
OPTICS COMMUNICATIONS
15 January 1983
TRANSVERSE EFFECTS IN ABSORPTIVE OPTICAL BISTABILITY J.V. MOLONEY, M. SARGENT III and H.M. GIBBS Optical Sciences Center, University o f Arizona, Tucson, A Z 85721, USA
Received 31 August 1982 Revised manuscript received 19 October 1982
Absorptive bistability loops are computed for a laser beam with a gaussian input spatial profile incident on a unidirectional ring cavity containing a saturable nonlinear medium. Sharp radially dependent switching with wide bistable loops is predicted for large Fresnel numbers. Intermediate Fresnel numbers result in whole beam switching with significant narrowing of the associated bistable loops. At low Fresnel numbers, bistability can disappear. Radially integrated hysteresis loops differ quantitatively from plane-wave predictions.
Absorptive optical bistability was the first theoretical model used to predict hysteresis in a passive optical cavity [1 ]. The experimental verification of this phenomenon, however, was observed first for a Fabry-Perot etalon in the dispersive limit [2]. Earlier and subsequent experiments sought absorptive bistability [2,3], but usually found dispersive bistability [2,3] *~ Theoretical modeling of these hysteresis phenomena has concentrated on plane-wave models [1,5,6] in many cases in the mean-field limit. In particular, Bonifacio and Lugiato [7] have analytically calculated the plane wave steady state solution for absorptive bistability in a ring cavity including propagation effects. Recent attempts have been made to model the more realistic case of a transverse spatial beam with transverse variations by assuming mode-matched, high-ffmesse cavities with negligible absorption per pass [8]. This situation differs substantially from the situation commonly encountered experimentally ~2 ,1 Absorptive optical bistability has been reported in ref. [4]. *2 A recent experiment by D.E. Grant and H.J. Kimble employs a high finesse confocal resonator to study absorptive bistability. This configuration allows strong diffraction coupling in the mode matched beam so that the cavity should support a single transverse mode. Good agreement is obtained with the theoretical mode-matched model of Drummond for small values of the cooperativity parameter C = aoL/2T although large discrepancies arise at large C values. Our system uses planar mirrors and no attempt is made to mode match the cavity (see ref. [4]). 0 030-4018/83/0000--0000/$ 03.00 © 1983 North-Holland
In the present paper we give a more realistic simulation of absorptive bistability. This work is motivated by our recent prediction of a radical departure from the results of these simple models in the case o f dispersive optical bistability [9]. For this latter case, radially-dependent nonlinear index changes can give rise to small-scale self-focusing instabilities or spatial ring structures in the transmitted profile. In the next section we make a quantitative comparison of absorptive bistability for a plane-wave and a gaussian transverse input. The transverse calculation is carried out for large and intermediate Fresnel numbers F (where F = nOw212/LL,~ n o is the medium index, Wl/2 the beam half width at half maximum, ~ the laser wavelength and 2L the cavity length). The main conclusions are that while the plane-wave and gaussian onaxis loops compare favorably at the high Fresnel number F = 8800 (the Uniform Plane Wave Approximation), the whole-beam (radially integrated) hysteresis shows much different behavior due to radially dependent switch-on. At the intermediate Fresnel number (F = 8.8) both on-axis and whole-beam hysteresis loops are barely discernible in marked contrast to the plane-wave case. At even lower Fresnel numbers the hysteresis disappears. The method o f solution has already been discussed in ref. [9]. Here we briefly outline the features appropriate for absorptive optical bistability. We consider the case of unidirectional ring cavity in the limit
289
Volume 44, number 4
OPTICS COMMUNICATIONS
that the atomic transverse and longitudinal relaxations are much faster than the cavity round-trip time. The wave equation describing the propagation of the laser beam through the nonlinear medium can be written
15 January 1983 PLANE WAVE F =
88
F =
8800
.....
ON-AXIS ON-AXIS
5.00
a)
as
O6(x,y,z)_ ~(z/2L)
[
1
c~0L 1 + t 6 1 2
i(ln 2V2)-]
"-~-°~ J6(x'y'z)(1)
where the first term on the right describes a two-level saturable absorber and the second is the diffraction term. c~0L is the linear absorption per pass, the complex field amplitude C is in units of the saturation amplitude C s and V2 is the transverse laplacian. Eq. (1) also describes free-space propagation if the twolevel-atom saturable term is removed (i.e., % = 0). The solution to eq. (1) is efficiently obtained by using a fast-Fourier-transform technique [9]. The beam is propagated through the nonlinear medium and around the cavity, added to the beam passing through the input mirror and fed back again until the transmitted beam reaches a steady state. The intensity reflection coefficient R is 0.9 for both the input and output mirrors for the present calculation. We choose the quantity aoL/T = 15, where T = 1 R, to ensure a wide plane-wave bistable loop +3 Our transverse calculations are restricted to one transverse dimension, i.e., the planar-waveguide case. The plane-wave solution is obtained by dropping the diffraction term from eq. (1) and numerically integrating the resulting equation. The plane-wave switch-up I t = 6.4 1s (I s = l fsl 2) and switch-down I , = 5 . 2 I s intensities can be read off directly from fig. l(a). The corresponding on-axis loop for high Fresnel number F = 8800 is shown in fig. l(b). The switch-up intensity has shifted to higher intensity (I t = 7Is) while the switch-down intensity is essentially the same. The higher switch-up intensity can be interpreted as the inability of the non-uniform intensity profile to saturate the medium efficiently. Otherwise both loops display the same qualitative features. Fig. l(c) shows the on-axis bistable loop for the intermediate Fresnel number case (/7 -- 8.8). The loop has narrowed dramatically, demonstrating the inability of *3 The plane-wave mean-field analytic model predicts hysteresis if o~oL/T> 8. The mean-field limit also requires that ~o L, T ,~ 1. 290
3.75
b
Iout
J
) ./'"
.c)
I i
1.25
4
5
6
7
8
'[in
Fig. 1. Comparison of plane wave and gaussian beam (onaxis) hysteresis loops. Parameters common to all calculations are c~0L = 1.5, T = 0.9, (a) plane wave hysteresis loop (b) on-
axis gaussian beam hysteresis with Fresnel number F = 8800 and (c) on-axis gaussian beam hysteresis with F = 8.8. Note that both low and high Fresnel number beam switch-on at approximately the same input intensity. All computations were limited to 200 cavity roundtrips and as a result critical slowing down causes the switch down discontinuity to be less pronounced in the plane wave case. the gaussian beam to hold the system in the on-state. (Note that I t = 7Is) is the same as the F = 8800 case (fig. l(b)) but the 1, = 6.8Is). This can be interpreted as being due to strong absorption of energy from the low-intensity wings of the strongly diffracted beam profile. Figs. 2(a) and 2(b) show the integrated whole beam hysteresis loops. These calculations have been limited to a maximum of 200 cavity roundtrips per input intensity for F = 8800 and F = 8.8, respectively. At high Fresnel numbers the switch-on is gradual, reflecting the radial dependence of the switching process. As a result the up-sweep and down-sweep portions of the high-transmission branch no longer coincide. The strong radial dependence is explicitly shown in the three-dimensional plot of fig. 3. The absence of index effects in the present situation avoids the small scale spatial oscillations encountered in the computations of ref. [9] *4 for dispersive bistability. Otherwise the radial dependent switching is qualitatively similar; the mechanism for radially de*4 See also ref. [10]. These authors assume a thin sheet nonlinearity and so avoid propagation effects.
Volume 44, number 4 F=
8.8
OPTICS COMMUNICATIONS
15 January 1983
.......
F = 8800
5.00 r
3.75
Iout 2.50
1.25 0
I
10
15
20
tin Fig. 2. Integrated whole beam hysteresis loops for Fresnel number (a) F = 8800 and (b) F = 8.8. The radially integrated intensities are defined as/in(out) = f0°° Iin(out) (x) dx.
Fig. 3. Three dimensional plots showing the strong radial dependence of the switching to the high transmission state at high Fresnel number (F = 8800). The lower plot shows the forward intensity sweep while the upper plot is the back sweep. For symmetry reasons we display only half of the transverse spatial profile.
....
o
0
"~
Fig. 4. Three dimensional plots showing whole beam switching at low Fresnel number. Note the significant narrowing of the hysteresis loop relative to fig. 3.
pendent switching involves gradual saturation o f the nonlinear medium. The whole-beam hysteresis loop for ( F = 8.8) again shows marginal bistability. This loop however retains the sharp switch-on feature indicative o f whole-beam switching. Three-dimensional plots corresponding to this case are shown in fig. 4. At lower Fresnel numbers, the hysteresis disappears, which may provide a clue to the reason why this type o f bistability has been difficult to observe experimentally. It is interesting to note that the F = 8.8 case corresponds to the Na experiment of Gibbs et al. in which they reported absorptive optical bistability with the laser tuned to a frequency having no nonlinear index (not on resonance because o f multiple transitions) and the intensity increased to saturate the medium [2,4]. The simulation shows a very narrow bistable loop consistent with the observations. To summarize our main results, we have studied absorptive optical bistability including transverse spatial beam variations and have compared the results with the corresponding plane-wave solutions. Medium propagation has been rigorously incorporated in the problem; numerical convergence required that we split the nonlinear medium up into 60 absorber sheets (see ref. [9]). At high Fresnel numbers, diffraction plays an insignificant role and switching to the high-transmission state occurs though a gradual 291
Volume 44, number 4
OPTICS COMMUNICATIONS
saturation of nonlinear medium. This very slow switchon, which is characteristic of absorptive bistable systems, manifests itself also in the radial switching. This is a further manifestation of critical slowing down in a higher dimension. Fig. 3 is particularly illuminating in this regard. Whole-beam hysteresis loops show less pronounced switch-on. At intermediate Fresnel numbers, whole-beam switching occurs, but bistability is substantially reduced. Indeed hysteresis disappears at lower Fresnel numbers. These behaviors show the inadequacy of the simple models, referred to in the introduction, as regards the quantitative behavior of absorptive bistable systems. The authors acknowledge support from the U.S. Air Force Office of Scientific Research and the U.S. Army Research Office.
References [1] A. SziSke, V. Daneu, J. Goldhar and N.A. Kurnit, Appl. Phys. Lett. 15 (1969) 376.
292
15 January 1983
[2] H.M. Gibbs, S.L. McCalland T.N.C. Venkatesan, Phys. Rev. Lett. 36 (1976) 1135. [3] H.M. Gibbs, S.L. McCall and T.N.C. Venkatesan, Optics News 5 (1979) 6. [4] W.J. Sandle and A. Gallagher, Phys. Rev. A 24 (1981) 2017; K.G. Weyer, H. Wiedenmann, M. Rateike, W.R. MacGillivray, P. Meystre and H. Walther, Optics Comm. 37 (1981) 426; D.E. Grant and H.J. Kimble, Optics Lett. 7 (1982) 353. [5] S.L. McCall, Phys. Rev. A 9 (1974) 1515. [6] R. Bonifacio and L.A. Lugiato, Phys. Rev. A 18 (1978) 1129. [7] R. Bonifacio and L.A. Lugiato, Lett. Nuovo Cimento 21 (1978) 505. [8] J.H. Marburger and F.S. Felber, Phys. Rev. A 17 (1981) 301; P.D. Drummond, IEEE J. Quantum Electron. 17 (1981) 301; R.J. Ballagh, J. Copper, M.W. Hamilton, W.J. Sandle and D.M. Warrington, Optics Comm. 37 (1981) 143. [9] J.V. Moloney and H.M. Gibbs, Phys. Rev. Lett. 48 (1982) 1607; J.V. Moloney, M.R. Belic and H.M. Gibbs, Optics Comm. 41 (1982) 379. [10] N.N. Rosanov and V.E. Semenov, Optics Comm. 38 (1981) 435.
OPTICS COMMUNICATIONS
15 January 1983
TRANSVERSE EFFECTS IN ABSORPTIVE OPTICAL BISTABILITY J.V. MOLONEY, M. SARGENT III and H.M. GIBBS Optical Sciences Center, University o f Arizona, Tucson, A Z 85721, USA
Received 31 August 1982 Revised manuscript received 19 October 1982
Absorptive bistability loops are computed for a laser beam with a gaussian input spatial profile incident on a unidirectional ring cavity containing a saturable nonlinear medium. Sharp radially dependent switching with wide bistable loops is predicted for large Fresnel numbers. Intermediate Fresnel numbers result in whole beam switching with significant narrowing of the associated bistable loops. At low Fresnel numbers, bistability can disappear. Radially integrated hysteresis loops differ quantitatively from plane-wave predictions.
Absorptive optical bistability was the first theoretical model used to predict hysteresis in a passive optical cavity [1 ]. The experimental verification of this phenomenon, however, was observed first for a Fabry-Perot etalon in the dispersive limit [2]. Earlier and subsequent experiments sought absorptive bistability [2,3], but usually found dispersive bistability [2,3] *~ Theoretical modeling of these hysteresis phenomena has concentrated on plane-wave models [1,5,6] in many cases in the mean-field limit. In particular, Bonifacio and Lugiato [7] have analytically calculated the plane wave steady state solution for absorptive bistability in a ring cavity including propagation effects. Recent attempts have been made to model the more realistic case of a transverse spatial beam with transverse variations by assuming mode-matched, high-ffmesse cavities with negligible absorption per pass [8]. This situation differs substantially from the situation commonly encountered experimentally ~2 ,1 Absorptive optical bistability has been reported in ref. [4]. *2 A recent experiment by D.E. Grant and H.J. Kimble employs a high finesse confocal resonator to study absorptive bistability. This configuration allows strong diffraction coupling in the mode matched beam so that the cavity should support a single transverse mode. Good agreement is obtained with the theoretical mode-matched model of Drummond for small values of the cooperativity parameter C = aoL/2T although large discrepancies arise at large C values. Our system uses planar mirrors and no attempt is made to mode match the cavity (see ref. [4]). 0 030-4018/83/0000--0000/$ 03.00 © 1983 North-Holland
In the present paper we give a more realistic simulation of absorptive bistability. This work is motivated by our recent prediction of a radical departure from the results of these simple models in the case o f dispersive optical bistability [9]. For this latter case, radially-dependent nonlinear index changes can give rise to small-scale self-focusing instabilities or spatial ring structures in the transmitted profile. In the next section we make a quantitative comparison of absorptive bistability for a plane-wave and a gaussian transverse input. The transverse calculation is carried out for large and intermediate Fresnel numbers F (where F = nOw212/LL,~ n o is the medium index, Wl/2 the beam half width at half maximum, ~ the laser wavelength and 2L the cavity length). The main conclusions are that while the plane-wave and gaussian onaxis loops compare favorably at the high Fresnel number F = 8800 (the Uniform Plane Wave Approximation), the whole-beam (radially integrated) hysteresis shows much different behavior due to radially dependent switch-on. At the intermediate Fresnel number (F = 8.8) both on-axis and whole-beam hysteresis loops are barely discernible in marked contrast to the plane-wave case. At even lower Fresnel numbers the hysteresis disappears. The method o f solution has already been discussed in ref. [9]. Here we briefly outline the features appropriate for absorptive optical bistability. We consider the case of unidirectional ring cavity in the limit
289
Volume 44, number 4
OPTICS COMMUNICATIONS
that the atomic transverse and longitudinal relaxations are much faster than the cavity round-trip time. The wave equation describing the propagation of the laser beam through the nonlinear medium can be written
15 January 1983 PLANE WAVE F =
88
F =
8800
.....
ON-AXIS ON-AXIS
5.00
a)
as
O6(x,y,z)_ ~(z/2L)
[
1
c~0L 1 + t 6 1 2
i(ln 2V2)-]
"-~-°~ J6(x'y'z)(1)
where the first term on the right describes a two-level saturable absorber and the second is the diffraction term. c~0L is the linear absorption per pass, the complex field amplitude C is in units of the saturation amplitude C s and V2 is the transverse laplacian. Eq. (1) also describes free-space propagation if the twolevel-atom saturable term is removed (i.e., % = 0). The solution to eq. (1) is efficiently obtained by using a fast-Fourier-transform technique [9]. The beam is propagated through the nonlinear medium and around the cavity, added to the beam passing through the input mirror and fed back again until the transmitted beam reaches a steady state. The intensity reflection coefficient R is 0.9 for both the input and output mirrors for the present calculation. We choose the quantity aoL/T = 15, where T = 1 R, to ensure a wide plane-wave bistable loop +3 Our transverse calculations are restricted to one transverse dimension, i.e., the planar-waveguide case. The plane-wave solution is obtained by dropping the diffraction term from eq. (1) and numerically integrating the resulting equation. The plane-wave switch-up I t = 6.4 1s (I s = l fsl 2) and switch-down I , = 5 . 2 I s intensities can be read off directly from fig. l(a). The corresponding on-axis loop for high Fresnel number F = 8800 is shown in fig. l(b). The switch-up intensity has shifted to higher intensity (I t = 7Is) while the switch-down intensity is essentially the same. The higher switch-up intensity can be interpreted as the inability of the non-uniform intensity profile to saturate the medium efficiently. Otherwise both loops display the same qualitative features. Fig. l(c) shows the on-axis bistable loop for the intermediate Fresnel number case (/7 -- 8.8). The loop has narrowed dramatically, demonstrating the inability of *3 The plane-wave mean-field analytic model predicts hysteresis if o~oL/T> 8. The mean-field limit also requires that ~o L, T ,~ 1. 290
3.75
b
Iout
J
) ./'"
.c)
I i
1.25
4
5
6
7
8
'[in
Fig. 1. Comparison of plane wave and gaussian beam (onaxis) hysteresis loops. Parameters common to all calculations are c~0L = 1.5, T = 0.9, (a) plane wave hysteresis loop (b) on-
axis gaussian beam hysteresis with Fresnel number F = 8800 and (c) on-axis gaussian beam hysteresis with F = 8.8. Note that both low and high Fresnel number beam switch-on at approximately the same input intensity. All computations were limited to 200 cavity roundtrips and as a result critical slowing down causes the switch down discontinuity to be less pronounced in the plane wave case. the gaussian beam to hold the system in the on-state. (Note that I t = 7Is) is the same as the F = 8800 case (fig. l(b)) but the 1, = 6.8Is). This can be interpreted as being due to strong absorption of energy from the low-intensity wings of the strongly diffracted beam profile. Figs. 2(a) and 2(b) show the integrated whole beam hysteresis loops. These calculations have been limited to a maximum of 200 cavity roundtrips per input intensity for F = 8800 and F = 8.8, respectively. At high Fresnel numbers the switch-on is gradual, reflecting the radial dependence of the switching process. As a result the up-sweep and down-sweep portions of the high-transmission branch no longer coincide. The strong radial dependence is explicitly shown in the three-dimensional plot of fig. 3. The absence of index effects in the present situation avoids the small scale spatial oscillations encountered in the computations of ref. [9] *4 for dispersive bistability. Otherwise the radial dependent switching is qualitatively similar; the mechanism for radially de*4 See also ref. [10]. These authors assume a thin sheet nonlinearity and so avoid propagation effects.
Volume 44, number 4 F=
8.8
OPTICS COMMUNICATIONS
15 January 1983
.......
F = 8800
5.00 r
3.75
Iout 2.50
1.25 0
I
10
15
20
tin Fig. 2. Integrated whole beam hysteresis loops for Fresnel number (a) F = 8800 and (b) F = 8.8. The radially integrated intensities are defined as/in(out) = f0°° Iin(out) (x) dx.
Fig. 3. Three dimensional plots showing the strong radial dependence of the switching to the high transmission state at high Fresnel number (F = 8800). The lower plot shows the forward intensity sweep while the upper plot is the back sweep. For symmetry reasons we display only half of the transverse spatial profile.
....
o
0
"~
Fig. 4. Three dimensional plots showing whole beam switching at low Fresnel number. Note the significant narrowing of the hysteresis loop relative to fig. 3.
pendent switching involves gradual saturation o f the nonlinear medium. The whole-beam hysteresis loop for ( F = 8.8) again shows marginal bistability. This loop however retains the sharp switch-on feature indicative o f whole-beam switching. Three-dimensional plots corresponding to this case are shown in fig. 4. At lower Fresnel numbers, the hysteresis disappears, which may provide a clue to the reason why this type o f bistability has been difficult to observe experimentally. It is interesting to note that the F = 8.8 case corresponds to the Na experiment of Gibbs et al. in which they reported absorptive optical bistability with the laser tuned to a frequency having no nonlinear index (not on resonance because o f multiple transitions) and the intensity increased to saturate the medium [2,4]. The simulation shows a very narrow bistable loop consistent with the observations. To summarize our main results, we have studied absorptive optical bistability including transverse spatial beam variations and have compared the results with the corresponding plane-wave solutions. Medium propagation has been rigorously incorporated in the problem; numerical convergence required that we split the nonlinear medium up into 60 absorber sheets (see ref. [9]). At high Fresnel numbers, diffraction plays an insignificant role and switching to the high-transmission state occurs though a gradual 291
Volume 44, number 4
OPTICS COMMUNICATIONS
saturation of nonlinear medium. This very slow switchon, which is characteristic of absorptive bistable systems, manifests itself also in the radial switching. This is a further manifestation of critical slowing down in a higher dimension. Fig. 3 is particularly illuminating in this regard. Whole-beam hysteresis loops show less pronounced switch-on. At intermediate Fresnel numbers, whole-beam switching occurs, but bistability is substantially reduced. Indeed hysteresis disappears at lower Fresnel numbers. These behaviors show the inadequacy of the simple models, referred to in the introduction, as regards the quantitative behavior of absorptive bistable systems. The authors acknowledge support from the U.S. Air Force Office of Scientific Research and the U.S. Army Research Office.
References [1] A. SziSke, V. Daneu, J. Goldhar and N.A. Kurnit, Appl. Phys. Lett. 15 (1969) 376.
292
15 January 1983
[2] H.M. Gibbs, S.L. McCalland T.N.C. Venkatesan, Phys. Rev. Lett. 36 (1976) 1135. [3] H.M. Gibbs, S.L. McCall and T.N.C. Venkatesan, Optics News 5 (1979) 6. [4] W.J. Sandle and A. Gallagher, Phys. Rev. A 24 (1981) 2017; K.G. Weyer, H. Wiedenmann, M. Rateike, W.R. MacGillivray, P. Meystre and H. Walther, Optics Comm. 37 (1981) 426; D.E. Grant and H.J. Kimble, Optics Lett. 7 (1982) 353. [5] S.L. McCall, Phys. Rev. A 9 (1974) 1515. [6] R. Bonifacio and L.A. Lugiato, Phys. Rev. A 18 (1978) 1129. [7] R. Bonifacio and L.A. Lugiato, Lett. Nuovo Cimento 21 (1978) 505. [8] J.H. Marburger and F.S. Felber, Phys. Rev. A 17 (1981) 301; P.D. Drummond, IEEE J. Quantum Electron. 17 (1981) 301; R.J. Ballagh, J. Copper, M.W. Hamilton, W.J. Sandle and D.M. Warrington, Optics Comm. 37 (1981) 143. [9] J.V. Moloney and H.M. Gibbs, Phys. Rev. Lett. 48 (1982) 1607; J.V. Moloney, M.R. Belic and H.M. Gibbs, Optics Comm. 41 (1982) 379. [10] N.N. Rosanov and V.E. Semenov, Optics Comm. 38 (1981) 435.