ON THE CAVITY OPTIMIZATION OF THE PHOTOREFRACTIVE BISTABLE ETALON
Tomislav A. Dzhekov Faculty of Electrical Engineering, Skopje, R. of Macedonia
Keywords: optical devices, resonator cavities, Fab^-Perot interferometer, photorefractive bistabiiity, pinotorefractive bistable etalons intrinsic
'pfpSChin^nrc^s	- -^^v optimLS?::;
^^^	' s Of a study on the problem of bistable cavity optimization (on transmission and on reflection) for
^^PH ir Tr % 'Tf	reflectivities, are presented. Three different optimization conditions have been considered (fixed fin^se
fixed end-face reflectivity and fixed absorption per pass). Special attention has been given to the case of high finesse cavity.
Optimizacija votline bistabilnega fotorefraktivnega etalona
optične, votline resonatorske, Fab^-Perot interferometri, bistabilnost fotorefraktivna, etaloni fotorefral<tivni blstabilni naprave bistabilne notranje, vot ne bistabilne, optimizacija votlin, optimizacija votlin za prenos, optimizacija votlin za refleksijo, optimizacij^ vo Hn
za dano absorpcijo optično, votline z visoko finostjo površine	vuuiii
Povzetek: V prispevku so predstavljeni analitični in numerični rezultati študije problema optimizacije votline fotorefraktivnega bistabilnega etalona (za transmisijo m refleksijo) za primer vzbujanja z ravnim valom pri enaki odbojnosti končnih zrcal
Prispevek obravnava tri različne optimlzacijske pogoje: fiksno finost votline, fiksno reflektivnost zrcal in fiksno absorpcijo na prehod Posebna pozornost je posvečena votlini z visoko finostjo.
1. INTRODUCTION
Optical bistabiiity has attracted a continuous interest over the past two decades /1 -18/. This is especially true for the intrinsic photorefractive bistabiiity (the nonlinear-ity of the medium of the photorefractive Fabry-Perot etalon is of the Kerr-type, i.e. the refractive index is given by:
n = ng + n2l = ng +
where no and n2 are the linear and nonlinear refractive indexes of the medium, I is the optical intensity, E is the optical electric field vector and iio is the wave impedance in the vacuum), owing to its potential application to all optical signal processing /10,11,18/.
The critical intensity Ic /6/ is certainly a parameter of central importance concerning the potential applications of the intrinsic photorefractive bistable etalons. Since lower critical intensity means switching at lower powers, it is desirable, of course, for this parameter to be as small as possible. Therefore, cavity optimization of the photorefractive bistable etalon implies such a choice of the final parameters used for fabrication (i.e., the end-face reflectivities and the absorption per pass), which leads to the smallest possible critical intensity for given conditions. Other important parameters are the transmission difference To (for the case when the transmitted wave is used as an output) and the reflection difference Rd (for the case when the reflected wave
is used as an output) /9/. It is necessary for these parameters to be as large as possible, because larger Td (or Rd) means in, principle, larger output signal.
In /6/, which is the first paper to treat this question, Miller reduces the problem of cavity optimization to minimization of/c for fixed finesse. He concentrates on transmission, but does not include explicitly the transmission difference in his analysis. One could hardly say, however, that the cavity is optimally designed if Ic was minimized at the expense of To (for the case of transmission) or at the expense of flo (for the case of reflection). Wherrett /9/ gives emphasis on reflectivity and is concerned with the dependence of the critical intensity and the reflection difference on the values of end-face reflectivities for specific absorption per pass conditions. No attempt is made, however, to obtain explicit expressions for the optimal values of the end-face reflectivities and of the absorption per pass for given conditions.
Normally, one expects from a cavity to have small Ic, but, at the same time, large Td (or Rd). Thus, it seems more meaningful if by cavity optimization is understood minimization of/c/To {or URo), rather than minimization of Ic. As can be easily seen, the former insures, effectively, maximization of the efficacy by which the input power is used. Therefore, in this paper we pay special attention to the minimizations oUc/Td and Ic/Rd. This is done for three different conditions, i.e. for given finesse O, for given end-face reflectivity R (we assume equal end-face reflectivities), and for given absorption per pass/\.
2. THEORETICAL OUTLINE
The following parameters characterize the analyzed bistable cavity:
R a d A B Ra F <t>
Intensity reflectivity of the faces Linear absorption coefficient Cavity length
1 - e'"*^ (absorption per pass)
e""*^ EE 1-A (absorption per pass complement)
Re'""^ = RB (effective intensity reflectivity)
4Ra/(1-Ra)^ = 4RB/(1-RB)2
/ 2 = uRa'^' / (1 -Ra) (cavity finesse)
In terms of these parameters, the critical intensity (Ic), the transmission difference (To) and the reflection difference (Rd) are given by /6, 9/:
(1)
ßH
where ß = Sna / is a constant which contains all the relevant material properties, and

16TC ^/2
(1-R){l-e-^^)(1 + R„.) H(F)
(1-Ra

(2)
with
H(F) = j(F + 2)[(F + 2)' +	- (F + 2)' - 2F
1/2
G(F) = 3(F + 2)-[(F + 2) +8F' is a figure of merit for the cavity design,
Td-

1-R^ e
2/„-ad
(3)
Rd=-
4 Re-"''(1-R)			
			2
(4)
B=1 and R = 1, except at the point P at which it becomes unlimited. On the other hand, Td and Rd are asymmetrical functions with respect to R=B. They are both equal to zero along B=0, R=0 and R=1 and equal to 4R / (H-R)2 at B =1. Along the diagonal OP, Td increases from zero to 1 /4, whereas Rd - from zero to 3/4. The point P is a singular point for these functions.
Fig. 1. Domain of definition of |i, Td and Rd
In the limit of high finesse (i.e., small A=1-B and small 1-R) is:

(1-R + 1-B)
»1
(5)
and expressions (2), (3) and (4) reduce to:
IXB-
(1-R)(1-B) 2 (1-R + 1-B)'
(1-Rf
Tn =
Rd =
° (1-R + 1-Bf (1-R)[1-R + 2(1-B)
(1-R + 1-B)
(6)
(7)
(8)
As can be seen, Td and Rd can be considered as functions of two variables - the mirrors' reflectivity R and the absorption per pass complement B = 1-A=e-«°. These functions are physically meaningful only within the domain 0<R<1, 0<B<1, Fig. 1. Note that m is a symmetrical function with respect to the line R=B, i.e. |i|b=x,r=y = m-Ib=y,r=x. It increases monotonically along the diagonal OP, from zero at point 0 to infinity at point P, being equal to zero along the lines B=0, R=0,
3. RESULTS AND DISCUSSION
A. Cavity optimization for given finesse
As was shown in /6/, the pairs of values of R and B which for given finesse F minimize the critical intensity Ic, are given by:
Table 1. Optimal values of R and B for given finesse <t>. / Ra= RB = {[1 + (n /	- n / 20
Minimized parameter
Ic
Location of the minimum
Kp - ^op = K
1/2
UTd
(optimization on
transmission)


1/2
B
R.
op(t)
^op(l)
Ic/Rd
(optimization on
reflection)
R
■op(r)
-2

P
1/2
cos
— arccos 3
P
3/2
+

P
1/2
COS
K
-- arccos
3
9
3/2
P )
+

for q <Q for q >0
B =
op(r) jrj
R.
op(r)
where:
q =-----^-^ 4- and p ■
216	24	2	^
36
F^op - Bop - f^o
In accordance with (1), (2), (3) and (4), minimizations of Ic/Td and Ic/Rd for given finesse <I> are equivalent to location of the maxima of the functions ft = (1-R)® B(1-B) andfr = (1-R)2 (1-B)(1-RaB) along the curve RB = Ra = const., respectively. Table 1 presents the required solutions. For completeness, the case of minimization of Ic is also included. Graphical presentations of these solutions are given in Fig. 2. Ro =Ra and Bo =1 are the pairs of values of R and B which for given finesse maximize To and Rd (the maximal value of To and Rd for given finesse is 4Ra / (1 +Ra)^). As one can see, cavity optimization on transmission requires considerably smaller values for R, and, therefore, larger values for B (thinner etalon), than are the values that minimize the critical intensity. The values of R for cavity optimization on reflection are larger then for cavity optimization on transmission, but, still smaller than the ones that minimize the critical intensity. For the special case of high finesse cavity, relations given in Table 1 reduce to those in Table 2.
fiq
Rop,Bop Rop(t) Bop(l) Rop(r) Bop(r)
0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8 25.6 51.2
0
Fig. 2. Pairs of optimal values of R and B, for given finesse O. Rop and Bop, Rop(t) and Bop(t), and Rop(r) and Sopw correspond to minimization of Ic, Ic/Td and Ic/Rd, respectively.
Table 2. (high finesse cavity) Optimal values of R and B for given finesse <5.
Minimized parameter
Location of the minimum
Ic

UTo
R.
3(1
B
op(t)
R
op{l)
4
URd
R
{i - vT?) + 32(1 + - (7 - Vn)
op(r)
B.
R.
2(1 + Vn) 1 + V17
1-
9 + V17

op{r)
R
op{r)
9 + Jri
(i-K)
To show that cavity optimization based on minimization of Ic/Td or on minimization of Ic/Rd could be advantageous or more acceptable than cavity optimization based on minimization of Ic, it is useful to calculate the values of Ic, To and Rd for each of the three cases. Such calculations have been done for a high finesse cavity, Table 3. Comparing the presented values, we note that a cavity optimized for a minimum Ic/Td is characterized by a 2.25 times larger transmission difference than a
cavity optimized for a minimum ic, and that this is paid by an 33% increase in the required holding power (critical intensity). Also, the optimization for a minimum Ic/Rd offers a 13% larger reflection difference for a 5% increase in the holding power, as compared to cavity optimization for minimum Ic- We also note that in each case Rd is considerably larger than Td. This clearly indicates that the reflection mode of operation can prove to be better suited for device purposes.
Table 3. Critical intensity, transmission difference and reflection difference of an optimized high finesse cavity (optimization for given finesse)*
Optimization criteria	Ic	Td	Rd	Ic/Td	Ic/Rd
minimum Ic	4.000(1	1/4	3/4	16.000 {l-Ra)Co	^5.333(l-RJCo
minimum Ic/Td		9/16	15/16		«5MS(l-RcJCo
minimum Ic/Rd	«4.202(1	«0.372	«0.848	«I1.296(l-i?«jC,	^.955(l-RJCo
is a constant which depends on the properties of the medium.
B. Cavity optimization for given absorption per pass
In accordance with (1), (2), (3) and (4), minimizations of Ic, Ic/Td and Ic/Rd for given absorption per pass A = 1 -B are equivalent to solving the equations / 3R = 0 and 3(|j,Rd) / 3R = 0, respectively. The results will be of the form R'op = R'op(B), R'op(t) = R'op{t)(B), and R'op{r) = R'op(r)(B), respectively, where R'op, R'op(t) and R'op(r) are the required optimal values of R. Because of the very complex dependence of )i on R and B (note that H(F) and G(F) are functions of R and B!), it is clear that analytical solutions of these equations are not possible. In Fig. 3, we present the solutions obtained by numerical methods. R'to and RVo are the of values of R which for given finesse maximize To and Rd, respectively. As expected, cavity optimization on transmission for given absorption per pass will lead to smaller values for R than cavity optimization on reflection. For B=i>0, R'op, R'op(t) and R'op(r) assume the values 1/2, 2/5 and 1/2, respec-
1 -		
0.9 -		
0.8		
0.7 -		jy^ ^^ Ji * '
0.6 -		


-rjK- -X
X
0.3 -
0.2 --
; j . -X- - K - - x
. -X- - x-

Fig. 3.
B
Optimal vaiues of R for given B. R'op, R'op(t) andR'op(r) correspond to minimizations ofic, Ic/Td and Ic/Rd, respectively.
tively, as obtained theoretically. For B=>0, R'op, R'op(t) and R'op(r) approach unity.
The case of high finesse cavity allows analytical treatment. The corresponding expressions of R'op, R'op(t) and R'op(r), obtained by using (6), (7) and (8), are given in Table 4.
Table 5 presents the calculated values of Ic, To and Rd for high finesse cavities, optimized for minimum Ic, for minimum Ic/Td or for minimum Iq/Rd.
Table 4. (high finesse cavity) Optimal values of R for given B.
minimized parameter	Location of the minimum
Ic	R' -1 K2
IJTD	
URd	
C. Cavity optimization for given end-face reflectivity
Minimizations of Ic, Ic/Td and Ic/Rd for given end-face reflectivity R are equivalent to solving the equations a|i/8B=0, f:)(|iTD)/aB=0 and a(|aRD)/r:)B=0, respectively. The results are of the form B'op=B'op(R), B'op(t) = B'op(t)(R), and B'op(r) = B'op(r)(R), respectively, where B'op, B'op(t) and B'op(r) are the required optimal values of B. As in the previous case, analytical solutions
Table 5. Critical intensity, transmission difference and reflection difference of an optimized high finesse cavity (optimization forgiven B= 1-A)
Optimization criteria	Ic	Td	Rd	Ic/Td	Ic/Rd
minimum Ic	6.750(l-B)Co	«0.111	«0.555	«60.756 (1-5;C„	«12.151(1-5;C„
minimum Ic/Td		0.360	0.840	«28.444(1	«12.405 {l-B)Co
minimum Ic/Rd	«7.452(1	«0.211	0.740	«35.318(1	«10.070(1-5;C„
of these equations are not possible. In Fig. 4, we present the results obtained by numerical methods. B'o=1 presents the values of B which, for given R, maximize To and Rd. As expected, for given end-face reflectivity R, the optimal values of B for minimum Ic/Td are larger (thinner cavity is required) than are the optimum values B for minimum Ic or for minimum Ic/Rd. One can show analytically, that for R=»0, B'op, B'op(t) and B'op(r) approach 1/2, 3/4 and 2/3, respectively.
Table 6. (high finesse cavity) Optimal values of B for given R.
R
Fig. 4. Optimal values of B for given R, B'op, ß'op{t) and B 'op(r) correspond to minimizations of Ic, Ic/Td and Ic/Rd, respectively.
It is easy to show, by using (6), (7) and (8), that for the case of high finesse cavity B'op=B'op(R), B'op{t)=B'op(t)(R), and B'op(r) = B'op(r)(R) reduce to the simple expressions given in Table 6.
Table 7 presents the calculated values of Ic, To and Rd for a high finesse cavity, optimized for minimum Ic, for minimum Ic/Td or for minimum Ic/Rd.
Minimized parameter	Location of the minimum
Ic	2
Ic/To	ß -1 ^opi,) - 1 4
Ic/Rd	o . , ! - R
4.	CONCLUSIONS
We have been able within the limitations of the plane-wave approximation to give criteria for optimizing the design of a refractive nonlinear Fabry-Perot etalon in the presence of linear absorption for minimum critical intensity, minimum critical intensity - transmission difference ratio or minimum critical intensity - reflection difference ratio. Three optimization conditions have been considered: fixed finesse, fixed absorption per pass and fixed end-face reflectivity.
5.	REFERENCES
/1/ H. M, Gibbs, S. L McCall and T. N. C, Vgncatesan, "Differential gain and bistability using a sodium-filled Fabry-Perot interferometer", Physical Review Letters, 36, 1135-1138 (1976).
/2/ J. H. Marburger and F. S. Falber, "Theory of lossless nonlinear Fabry-Perot interferometer", Physical Review, A-17, 335-342 (1978)
/31 H. M. Gibbs, S. L. McCall, T. N. C. Vencatesan, A. 0. Gossard, A. Passner and W. Wiegmann, "Optical bistability in semiconductors", Applied Physics Letters, 35, 451-453 (1979),
Table 7. Critical intensity, transmission difference and reflection difference of an optimized high finesse cavity (optimization for given end-face reflectivity R)
Optimization criteria	Ic	Td	Rd	IC/Td	Ic/Rd
minimum Ic	6.750(l-i?)Co	«0.444	«0.888	«15.803 {\-R)Co	«7.601(l-i?;Co
minimum Ic/To		0.640	0.960	«12.207(l-i?;C„	«8138(l~i?;Co
minimum Ic/Rd	«6.8461(1	«0.504	«0.916	«13.5 73(1-WC;,	« 7.468(1
/4/ T, Bischolbergerand Y. R. Shen, "Theoretical and experimental study of the dynamic behavior of a non-linear Fabry-Perot interferometer", Physics Review, A-19, 1169-1176 (1979).
/5/ P. W. Smith, "Hybrid bistable optical devices". Optical Engineering, 19(4), 456-462 (1980).
/6/ D. A. B. Miller, "Refractive Fabry-Perot bistability with linear absorption: Theory of operation and cavity optimization", IEEE Journal of Quantum Electronics,Vol. 17(3), 306-311 (1981).
/7/ ,D. A. B. Miller, S. D. Smith and C. T. Seaton, "Optical bistability in semiconductors", IEEE Journal of Quantum Electronics, Vol. 17(3), 312-317 (1981).
/8,' W. J. Firth and E. M. Wright, "Theory of Gaussian-beam optical bistability", Optical communications. Vol. 40, 233 (1982).
/9/ B. S. Wherrett, "Fabry-Perot bistable cavity optimization on reflection", IEEE Journal of Quantum Electronics, Vol. 20, 646-651 (1984).
/10/ S. D. Smith, A. C. Walker, B. S. Wherrett, F. A. P. Tooley, J. G. H. Mathew, M. R. Taghizadeh and I. Janossy, "Cascadable digital optical logic circuit elements in the visible and infrared: demonstrations of some first all-optical circuits", Applied Optics, 25, 1586-1593 (1986).
/11/ M. E. Warren, S. W. Koch and H. M. Gibbs, "Optical bistability, logic gating and waveguide operation in semiconductor eta-Ions", Computer, University of Arizona, 68-81, December 1987,
/12/ M. Haelterrnan, M. D. Tolley and G. Vitrant, "Transverse effects in optical bistability with nonlinear Fabry-Perot etalon: a new theoretical approach", Journal of Applied Physics, 67(6), 2725-2730 (1990).
/13/ R. Reinish and G. Vitrant, "Transferse effects and enhanced dispersive optical bistability in a nonlinear Fabry-Perot filled with self-focusing medium". Journal of Applied Physics, 67(11), 6671-6674 (1990).
/14/ F. J. Fraile-Pelaez, J Capmany and M. A. Muhel, "Transmission bistability in a double-coupler fiber ring rezonator". Optical Letters, Vol. 16, 907-909 (1991)
/15/K. Ogusu and S. Yamamoto, "Nonlinear fiber Fabry-Perot resonator using termooptic effect", Journal of Lightwave Technology, Vol. 11, 1774-1781 (1993)
/16/J. Capmany, F. J. Fraile-Pelaez and M. A. Muriel, "Optical bistability and differential amplification in nonlinear fiber resonators", IEEE Journal of Quantum Electronics,Vol 30 2578-2588 (1994).
/17/ K. Ogustu, H. Shigekuni and Y. Yokota, "Dynamic transmission properties of a nonlinear ring resonator", "Optical Letters Vol. 20, 2288-2290 (1995),
/18/ K, Ogustu, "Dynamic behavior of reflection optical bistability in a nonlinear ring resonator", IEEE Journal of Quantum Electronics, Vol, 32, 1537-1543 (1996),
dr. TomislavA. Dzhekov Faculty of Electrical Engineering Cyril and Methody University Orce NIkolovbb., Skopje Republic of Macedonia Tel.: (+389 91) 363566 Fax.: (+389 91) 364 262, E-mail: tdzhekov ©cerera.etf.ukim.edu.mk
Prispelo (Arrived):17.01.1997 Sprejeto (Accepted):06.05.1997