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THEORY OF THE NONLINEAR OPTICAL RESPONSE OF ACTIVE MULTILAYER
INTERFERENCE FILTERS
D. Hutchings, B. Wherrett, D. Frank
To cite this version:
D. Hutchings, B. Wherrett, D. Frank. THEORY OF THE NONLINEAR OPTICAL RESPONSE
OF ACTIVE MULTILAYER INTERFERENCE FILTERS. Journal de Physique Colloques, 1988, 49
(C2), pp.C2-119-C2-122. �10.1051/jphyscol:1988227�. �jpa-00227644�
JOURNAL DE PHYSIQUE
Colloque C2, Suppl6ment au n06, Tome 49, juin 1988
THEORY OF THE NONLINEAR OPTICAL RESPONSE OF ACTIVE MULTILAYER INTERFERENCE FILTERS
D.C. HUTCHINGS, B.S. WHERRETT and D. FRANK*
Department of Physics, Heriot-Watt University, Riccarton, GB-Edinburgh EH14 4AS, Scotland, Great-Britain
~ n s t i t u t fiir Theoretische Physik, Aachen, F. R. G.
Abstract
-
A general theory for the response of filters containing layers of nonlinear refractive material is presented. For specific filter designs, optimisation for low-power switching is addressed.1 - INTRODUCTION
Intrinsic refractive optical bistability has now been observed in many semiconductors /I/.
Switching power levels in such systems are typically in the range 1-100 mW. Potentially, such systems have applications as displays. spatial light modulators, optical image processors and optical computational elements, taking advantage of the two-dimensional parallel processing possible using optics. However, in order to realise this parallelism at moderate total power levels for an array of bistable elements, it will be necessary for the switching power of a single element to be less than 100 pW.
Refractive changes in semiconductors used in optical bistability are induced as a result of excitations generated by intense light levels. To date such excitations have been either electronic (carriers excited to higher energy states) as in InSb /1/ or thermal as in ZnSe /2/. Usually such refractive index changes are small and it is necessary to introduce some form of optical feedback in order to obtain a useful change in the output signal. In this paper various configurations which provide optical feedback are examined and the optimisation for low switching power levels is addressed.
2
-
OPTICAL BISTABILITY WITH A THERMAL EXCITATIONThin-film dielectric interference fiters have been shown to exhibit highly nonlinear transmission and reflection characteristics under conditions of cw laser irradiation. In particular, transphasor action and optical bistability have been observed in
narrow-bandpass Fabry-Perot filters containing ZnSe and/or ZnS layers /2-4/.
In a thin-film filter structure, there is little longitudinal variation in temperature and hence the optical nonlinearity can be treated as nonlocal. The change in the optical phase for a particular layer with thermo-optic coefficient an/aT, can be written / 5 / ,
Here D is the thickness of the layer, h the radiation wavelength in vacuum, A the absorptance function of the filter. Po the incident laser power and aT/BPA is the average on-axis temperature rise in the layer per unit power absorbed which depends only on the laser spot and the physical construction of the filter (for Gaussian beams with w > D, where w is the l/el spot radius, aT/aPA = (rZiiw~~)-l where K, is the
substrate conductivity).
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988227
C2-120 JOURNAL
DE
PHYSIQUEThis self-consistent equation is completely general and can be used to generate the nonlinear characteristics for any thermo-optic filter. It is only necessary to know the absorptance function for the filter. This is known analytically for a simple Pabry-Perot cavity and can be calculated numerically for more complicated structures, e.g. by the matrix method which equates the tangential components of the electric and magnetic fields across each interface /6/.
The nonlinear response can be obtained from equation (1) either in parametric form by calculating the input power consistent with each A+ /7/ or by a graphical method by considering the intersect of the absorptance function with a straight line with gradient inversely proportional to the incident power level, in a manner similar to that of Marburger and Felber / 8 / .
Switching levels in such devices are characterised by the critical switching power PC which marks the onset of bistability, i.e. bistability can only be obtained for P > PC irrespective of the initial detuning,
+,.
PC can be calculated analytically for some simple examples.For a simple Fabry-Perot filter it can be shown that for a fixed pair of mirror reflectivities (Rf.Rb) there is a particular value for the cavity length which minimises the critical switching power (Fig. l(a)) / 9 / . The absorption (aD) must be sufficient to raise the sample temperature, but not so much that the cavity finesse is reduced; at the optimum cd) = (2-Rf-Rb). The switching power at the minimum is essentially independent of cavity finesse (in contrast to the case for optoelectronic switches), because the excitation sink (heat diffusion) is predominantly longitudinal into the substrate whereas the equivalent sink in the electronic case is carrier recombination that occurs throughout the volume of the active material.
CAVITY LENGTH (OD)
LOG 10
CAVITY LENGTH ( a D )
d l , , , ,
a
O-4 -3 - 2 -1 0
LOG 10 CAVITY LENGTH (dl)
Fig. 1. Relative switching powers as a function of the dimensionless cavity absorption in themo-optic devices for the mirror reflectivities indicated. (a) Simple Fabry-Perot cavity. (b) Fabry-Perot cavity with thick metal rear-face mirror, (c) Asymnetric Double Fabry-Perot cavity where only the rear cavity is nonlinear and the front cavity and front mirror have been optimised for high contrast and low switch power.
One possible solution to this problem is to place the absorption outside of the cavity by using metal mirrors /10,11/. In cavity configurations where the front mirror is metallic, optical phase changes at the metal mirror can result in an asymmetric absorptance function which leads to unusual bistable switches (we have predicted and observed flbutterfly"
bistability which consists of two switch-downs). The lowest switching levels are obtained with a dielectric front mirror (non-absorbing) and a metallic rear mirror, for which conventional bistable loops are regained. In such a system there is no fundamental limit to the switching power, being only limited by the reflectivity obtainable from a metal mirror (Figure l(b)). To date, the lowest switching power so observed is 20 pW /12/.
using K15 liquid crystal material in a dielectridmetal cavity.
Multicavity filters have, in general, steeper transmission edges in comparison to single FP filters and therefore it may be possible to reduce switching levels in such devices.
In order to investigate this we have analysed two double cavity examples. For a sywnetric double FP cavity, it has been shown by the numerical model discussed earlier that the optimal reflectivity for the middle mirror is unity which is equivalent to a single FP cavity with a high reflection back-face mirror. This result reflects the fact that the extra absorption required for bistability in the second cavity reduces the steepness of the absorptance edge. We have also analysed an example of separating the absorbing layer and the thermo-optic layer by considering an asymmetric double FP which is absorbing in the front cavity and nonlinear in the rear. This device uses the optical phase change on reflection from the second cavity. when it is close to resonance, to switch the front cavity onto resonance. However, we conclude that the optimal reflectivity for the middle mirror approaches zero (Figure l(c)) which again results in a single cavity design. It can therefore be concluded that there is no significant decrease in switching powers for these multicavity designs.
3 - OPTICAL BISTABILITY WITH AN ELECTRONIC NONLINEARITY
In contrast to the thermal nonlinearity, the excitation sink is recombination rather than diffusion and hence there is no coupling between individual layers, i.e. the nonlinearity is local. In a similar manner to the thermal nonlinearity. the nonlinear optical
characteristics for any thin-film interference filter can be obtained by the use of a self-consistent equation,
A$j =
-
OnT aDj Ij(A$j)Kc (2)
where Aqj is the change in optical phase in the jth layer, on the refractive
cross-section, T the recombination time, a the absorption coefficient and Ij the average internal irradiance in the jth layer. However, unlike the thermal case, the dependence is on the internal irradiance in a particular layer, rather than some quantity in common for the entire filter, thus the calculation needs to be iterated until convergence is
reached.
For a simple Fabry-Perot, for a fixed pair of mirror reflectivities, the critical power is minimised for aD = (2-Rf-Rb)/4. In this case the value of the critical power at this minimum is reduced by increasing the cavity finesse (Fig. 2) /13/. Hence, thin high finesse filters are required.
However, as the sample thickness is reduced, surface recombination begins to play an important role /14,15/. For cavity lengths less than the diffusion length D
<
Ld, the effective recombination time becomes,where s is the surface recombination velocity which depends on the material and surface preparation. Hence for thin samples, the effective recombination time is proportional to the cavity length and an absolute lower limit is placed on the critical power as in the thermal case (Fig. l(a)).
It has been previously demonstrated that refractive changes are a similar order of magnitude in GaAs/AlGaAs quantum well material and bulk GaAs /16.17/. However in MQW1s, and other epitaxially grown structures, the carriers are confined thus making the surface recombination velocity s small, and lower switching powers should be achievable in optimised systems.
JOURNAL DE PHYSIQUE
Cavity length ,
a!D
Fig. 2. Relative switching power in a simple Fabry-Perot cavity with an electronic nonlinearity for the equal end-face reflectivities indicated.
In conclusion, we have presented self-consistent equations for calculating the nonlinear optical response of interference filters both for a thermal (non-local) and an electronic
(local) nonlinearity. These models have been used to minimise switching levels and the lowest powers are predicted in a Fabry-Perot cavity with a back-face absorber in the case of a thermal nonlinearity and in materials with a low surface recombination velocity in the case of an electronic nonlinearity.
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