HAL Id: jpa-00227609
https://hal.archives-ouvertes.fr/jpa-00227609
Submitted on 1 Jan 1988
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
OPERATION CHARACTERISTICS OF PIXELLATED NONLINEAR INTERFERENCE FILTERS
Erika Abraham, C. Godsalve, B. Wherrett
To cite this version:
Erika Abraham, C. Godsalve, B. Wherrett. OPERATION CHARACTERISTICS OF PIXELLATED
NONLINEAR INTERFERENCE FILTERS. Journal de Physique Colloques, 1988, 49 (C2), pp.C2-
43-C2-46. �10.1051/jphyscol:1988210�. �jpa-00227609�
OPERATION CHARACTERISTICS OF PIXELLATED NONLINEAR INTERFERENCE FILTERS
E. ABRAHAM, C. GODSALVE and B.S. WHERRETT
Department of Physics, Heriot-Watt University, Riccarton, GB-Edinburgh EH14 4AS, Scotland, Great-Britain
Abstract
-
We study the effect of material pixellation on the operatingcharacteristics of bistable elements using nonlinear interference filters. Packing densities of 104-105 per cm2 and 100 pW power levels are predicted. Recovery times are shown to be < 100 ps.
1 - INTRODUCTION
Interference filters used in optical bistability experiments can be modelled as high-finesse Fabry-Perot interferometers containing a thermal nonlinearity. It can be shown /1/ that when irradiated by a Gaussian beam of radius w and input power P at the appropriate (visible) wavelength, the change in film temperature AT G T-To at the beam centre is
where To is the ambient temperature, Ks is the thermal conductivity of the substrate and A(T) is the absorptance given by /1/
A(T) = Ao/[l
+
~(e-a(T,) - bAT)zl (2)where 0 is the angle of incidence of the beam and A,, G, a(T,), b are determined from experimental data: we use A, = 0.45, G = 0.18, a = 30.5, b = 0.055 and 0 = 40' throughout. Both the temperature and the transmission have the same threshold powers /I/.
The small volume where the absorption takes place is a source of heat (per unit time per unit volume) Q(L) for the heat equation, which in the steady state reads
VzT(r) =
-
Q(r)/K (3)The formal solution of (3) is
and, as in electrostatics, one can make a "multipole" expansion
where q =
S ,
d31' Q(f8) is the total heat produced per unit time. The l/r dependence of-
r(5) accounts for the long-range effect of heat sources. This argument also explains the
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988210
C2-44 JOURNAL DE PHYSIQUE
increasing separation for independent operation of optically-generated pixels with increasing number of beams /2/. Calculations show that in a uniform area of 4 cm2 an array of only 7 x 7 elements would operate independently on a glass substrate. Hence cross talk severely limits optical pixel densities in thermo-optic devices.
2 - MATERIAL PIXELLATION: SINGLE-PIXEL THEORY
In order to suppress cross-talk effects a physical division of the active region into discrete areas is necessary: this is what we define as material pixellation. To treat the problem theoretically we consider a cyclindrical rod as in Fig. 1. Then the problem reduces to solving (3) in two different regions and matching the solutions at the
interface (z = 0) through the boundary conditions. The active film is located at z = -1, with P being the length of the rod. We assume no heat loss from either the front surfaces or the lateral surface of the pixel. The solutipn in the pixel (referred to by the subscript 'p') reads /3/
substrate
70
€a UNPIXELLATEO
(GLASS SUBSTRATE)
50
- 40 6
30
m
t :
X) GLASS'GLASS GLASSISAPPHIRE 2 - - - - --
- - - - - - - - -I I I I I , ,
c 0 2 4 6 6 10
Fig. 1. Schemat:ic of pixel/substrate geometry. Fig. 2. Critical separation vs. holding power for an array of nine pixels
(1 = 4 W = 40 p ) .
The B's are determined from the boundary conditions at z = -2
-5
aTp/az = I A(T~)and at z = 0
TP = Ts (8)
In (6) J, is the zeroth-order Bessel function and
Em
is the mth zero of the first order Bessel function J,; in (7) I E P/m2 is the input intensity. The solution in the substrate (referred to by the subscript 'st) taken as a semi-infinite solid is /3/where p F r/w and Q, 5 -K, aTs/az is the flux at z = 0. Then using (6)-(10) an
infinite set of equations for the coefficients is obtained which can be safely truncated after the first. M = 25. Finally we obtain the film temperature T at r = o.
where i-3 is the figure of merit of the pixel. A comparison between (1) and (12) indicates that to achieve the same change in refractive index, i.e. the same AT, the relationship between power levels is
This expression is meaningful if the same w's and substrates are compared. In this case,
B
can be made large by choosing as pixel material a much poorer conductor than the substrate or by deep pixellation or indeed both.If i-3 >> 1 we can recast (12) as
where RTH : IL/ is the thermal resistance of the pixel. Clearly AT becomes
independent of
"e
he substrate's thermal conductivity. A consequence of this insulation effect is a negligible increase of the substrate's temperature. To see this, we note first that by dropping the summation of (11) the problem is reduced to a one-dimensional heat flow. The temperature at any point z in the pixel then is,Hence using (12) Tp (0) -To 1
= - < < 1 T-To B
It follows that a large i-3 suppresses cross talk effects since practically the temperature drop occurs along the pixel. For example if one takes a glass pixel
(Kglass = 0.011 W/cm°C) and a sapphire substrate (Ksapp = 0.19 W/OC cm), for IL = 40 pm, w = 10 pm, we obtain i-3 = 79. For the absorptance profile we adopted
T-To = 220°C for the upper branch, hence the temperature change at the interface is % 3OC according to (15)
-
and this has been checked numerically to be correct. Finally we note in (13) that the power P a w2 as opposed to P a w in the unpixellated sample.As it is well established in nonlinear devices, there exists a power/speed trade-off. For large
p,
Ts % To and then the recovery (cooling) time can be obtained from a simpleC2-46 JOURNAL DE PHYSIQUE
one-dimensional model which yields a quasi-exponential decay with a time constant
where Pp and Cp are, respectively. the density and specific heat of the pixel. This time constant is in agreement with numerical solutions of the full time-dependent heat equation /4/. In a glass pixel of 1 = 40 pm we obtain ro = 1.3 ms. Note that a characteristic energy Pro o: 1 is reduced weakly as 1 is reduced. Therefore the optimum apears to remain a thin, highly insulating layer on a good heat-sink substrate.
For an optimised filter we can calculate the critical power for bistability /5/
h a K , w
with h : vacuum wavelength, a : absorption coefficient of the filter's spacer and dn/dT : thermo-optic coefficient of refractive index. If we choose polyimide (Kpoly = 0.0017 W/OC cm) on sapphire and A = 514 an, a = 400 cm-I, w = 3 p, II = 3 p,
dn/dT = 2 x 10-4 OC-I we obtain f% = 127 and PC = 119 pX. Using these parameters in (16) with ppol 1.4 g cm-', C oly = 1.6 J g - I "C-1, we obtain from (16) ro = 47 ps.
The sntc~i;g energy will ge of the order of PC ro, i.e. 5.6 nJ.
3
-
PACKING DEN=The theory can be extended to an array of pixels to estimate the minimum interpixel separation for independent operation. Consider two pixels El and E2 initially in the OFF state whose film temperatures are TI and T,. If E, is switched on, the temperature increase at the interface of E, will result in the same temperature change in El itself since no heat loss is assumed. Furthermore if N pixels from a square array of 'lattice constant' d of ,which N-1 are in the ON state and the one at the centre, say El, is OFF then according to ref. /6/ only one equation has to be considered, namely /3/
As rlj a d, the problem of finding the minimum separation is to determine a critical d, dcrit, such that E, can sustain itself in the OFF state while all the other pixels are ON. In Fig. 2 we show plots obtained numerically of dcrit, for an array of nine elements, as a function of holding power PH (Pj PH for every j). The latter is normalised to the switch-up power PUP of the corresponding single pixel.
Extrapolation of the glass/sapphire curve by placing each pixel in a 40 x 40 gives a conservative packing density of 250 x 250 cm-2 for PH
-
0.85 Pup.4
-
CONCLUSIONSOur calculation:; predict that material pixellation of interference filters could lead to a substantial reduction of the switch powers (x 100 pW) and packing densities
>
lo4 per cmz.This would require < 10 W/cmZ heat sinking which is within current technical capability.
Assuming smaller 1 p-dimension pixels and a cycle of 5 ps, switch rates of 1010 gate Hz ~ mwould look possible. - ~
REFERENCES
/1/ Janossy, I., Taghizadeh, M.R., Mathew, J.G.H. and Smith, S.D., IEEE J. Quantum Electron., QE-21 1447 (1985).
/2/ Abraham, E., Opt. Lett.,
2,
689 (1986)./3/ Abraham, E., Godsalve, C. and Wherrett, B.S., J. Appl. Phys. (June 1988) to appear.
/4/ Abraham, E., Kar, A.K., Suttie, M.R., Harris, R.M., Walker, A.C. and Smith, S.D., Appl. Phys. Lett. (submitted).
/5/ Wherrett, :B.S., Hutchings, D. and Russell, D., J. Opt. Soc. Am., B3, 351 (1986).
/6/ Abraham, E. and Rae. C., J. Opt. Soc. A m . , B4, 490 (1987).
