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Two-photon optical bistability with spatial effects
Volume 79A, number 2,3
PHYSICS LETTERS
where z~is the atomic detuning. The quantities kea = k~~(S~ç + Sk), kbb = kab(S~ Sk) are linear susceptibilities at frequency wk for levels a, b, respectively. At steady state we find 2Ix 2)~’, (7) D=N(1 ÷62)(l +62 + Ix 1~ 2I P = N(y 12(1 + i6)x~x 11/71)’ (8) X (1+62 +Jx 21x 2)--1, 11 21 where Xk = X1/2(~yj)1/4Ek are the fields appropriately scaled. In single-photon OB the mean-field limit has been defined [11] ,and examined extensively [12,13] ; we adopt essentially the same definition here. With mirror transmissivity T(0 < T< 1), we set --
g
~N(y
1I2c~
0,
T~0. (9)
11/71)
C 2gL/T-+ constant, Cbeing the cooperativity parameter and L the length of the medium, and also impose the conditions for transmitted fields (xkT) and incident fields (xkI) __
IXkTI ~ 0,
IXkl~ 0;
29 September 1980
[12]). Eqs. (9), (11), (12) and (13) can now heapplied, whereupon the coupled equations of state 2 ~ [i
=
+
— —
1 +62÷X~~l
+
[C[X76 L
+
(y~/y)”2S~(1+ 62)] ~k] 1+6 2 +x~x~
2
(14)
1, 2 (1 k), are obtained, with Yk = xkl~/T1/2,Xk = xkTI/T”2 and ~tk = Xk nNLC~ISj~.En the two-photon Raman process, the right-hand sides of eqs. (1) and (2) nmst exhibit opposing signs; this leads again to eq. (14) with, however, a negative sign preceding C. Conditions (11) and (12) apply more generally when T is no longer small, with the result k, 1
=
=
T~2{Ixk(O)12
+
R2X~ (15)
2RIxk(O)IXkcos[~k + ~ ~ where ~pk(z)are the phases of the fields Ek(z), and R = I T. It remains to evaluate xk(0) in terms of Xk --
~‘
(10) xkTI/T112
-÷
const.,
IxkII/T”2
-+
const.
The subsequent development is analogous to that specified for one-photon OB by Carmichael and Herniann [12], thus we have the boundary conditions Xk(L)
=
xkT/T’12,
and ~k and for this purpose we separate the field equations into real and imaginary parts, thus: dIxk~= —2g dz 1
--
(11)
x 26 +eSk(l +62) + x 21x 2 11 21
d~k T~2xkIei0kT=xk(0)(1 ~~T)xk(L)e2iokRe1Kk(L+L.),
where 0kT’ 0kR are the phase changes attending transmission and reflection, andL’ is the cavity path external to the active medium. We conveniently set ~k2~
—2g ~ 11 ~
~
(16a)
(16b)
~
with e = (y~/y~)1I2 and = + 2gcS~z.Eqs. (l6a) imply the spatial conservation law (Manley—Rowe relationship) for absorption:
[Kk(L +L’)+2OkR] mod 2n’
and assume
1x 2 1x 2 = 1x 2 x 2. (17) For1(z)1 the Raman 2(z)1 process the 1(0)1 conservation 2(0)1law is similar, with addition signs replacing the subtraction signs in (17). The integrals of eqs. (16a) lead to —
__
0; Xk = (1 T)T~’~k const. (13) With the time derivatives in eqs. (1), (2) set to zero, and substituting the stationary values (7) and (8) of P and D, the two field equations can be integrated over space. All quantities F(z) are replaced by ~ = L1 —
-~
X f~F(z) dz, and fields ~k can be replaced with xk(L) since the mean-field limit implies linearity (see, e.g. ref. 154
IxkHxlI 1x 2~x 2 1 2~
+ 62 +
—
—CT= 4Ix
2 Ixi(0)I2) 1(L)~ x 2(L)x1 (0) :L 6 2(a) d/l
±
ln
~
--
+ ~
(1 8a)
Volume 79A, number 2,3
PHYSICS LETTERS
with ~ = 1x1(z) 2, j3~= x1 (0)12 T 1x2(0)I 2, and i = 1 for double absorption (upper sign) and i = 2 for the Raman case (lower sign). Also eqs. (16b) produce zL
f
—
(18b)
where ~3’= Ixk(z)I 2~ The logarithmic terms, as well as the Stark terms, make the analysis very difficult; where the Stark terms and detunings may be ignored, however, the phases ~k(z) can be set to zero, with the consequence that eq. (15) becomes xk(0) = TYk + RXk. (19)
are possible. We hope to pursue the analysis further in order to ascertain the effects of the Stark terms upon the switching processes. These calculations, and also numerical results, will be reported elsewhere. Future work is also expected to deal with the more general stability condition. One of us (JAH) wishes to acknowledge financial
The conservation identities evaluated at z = L are now
support from the Science Research Council. Helpful comments were Dr. G. New, Dr. J. Elgin, Dr. P. Knight andgiven Dr. by H. Carmichael.
+
z0
d~’{6~’)/~’
As the Stark terms have been ignored here, the equation of state is 2 —2Rp1/2cos(~ ~6lnp)] l/2~ Y XT~ [p +R (24) For full transmissivity (T = 1) we have Y = Xp1/2. Although the essential features of two-photon bistable behaviour are present in eq. (24), it [2] that large effects attributable tohas the been Starkshown terms
—
~k(0) = ~
29 September 1980
eSk [1 + 62(~’)1~3’~31
—
~0~’
~ X~= (TY~+RX 2 ~ (TV 2, 1) 2 +RX2) or alternatively
(20)
References
(Y 2 —X2)[Y2 +X2(1 +R)/T] (20’) =
±(Y1 X1)[Y1 +X1(1 +R)/T] —
.
As in ref. [2] we examine the effect of varying y1 ,
with Y2 held constant. Since V1 ~X1 we see that in double absorption resonance V2 ~ X2, and in Raman resonance Y2 ~ X2, thus the switching processes are quite different for each case. With the degenerate case of double absorption, in which only one field is involved, the integration of eqs. (16a) yields (dropping all k indices and redefining kab, so that C is changed by a factor of one half) —2CT = Ix(L) 12
—
Ix(0)I 2
—
Ix(L)I —2
z=L
+
J’
~—2~2~
+
Ix(0)I —2 (21)
d13.
z0
The absence of logarithmic terms makes possible an exact analytic solution. In particular, when ~ is a constant, eq. (21) rearranges to 2p(X, 6), (22) Ix(0)I 2 = X 2p(X,6) 1 +2CTX2—(l +62)X4 +
{[1 +2CTX2 —(1 ÷62)X4]2
Eli
F.T. Arecchi (1978) 65. and A. Politi, Lett. Nuovo Cimento 23
[21G.P.
Agrawal and C. Flytzanis, Phys. Rev. Lett. 44 (1980) 1058. [3] R. Bonifacio and L.A. Lugiato, Lett. Nuovo Cimento 21 (1978) 505, 510, 517. [4] R. Bonifacio, M. Gronchi and L.A. Lugiato, Nuovo Cimento (1979). [5] R. Bonjfacio, M. Gronchi and L.A. Lugiato, Theory of optical bistability, in: Laser spectroscopy IV, Proc. 4th Conf. on Laser spectroscopy (1979), eds. Walther and Rotlie (Springer, 1979). [6] S.S. Hassan, P.D. Drummond and D.F. Walls, Opt. Commun. 27 (1978) 480. [71M. Takatsuji, Phys. Rev. Al 1(1975) 619. [81 R.G. Brewer and E.L. Hahn, Phys. Rev. All (1975) 1641. [9] N. Dutta, Phys. Lett. 69A (1978) 21. [101 L.M. Narducci, W.W. Eidson, P. Furcinitti and D.C. Eteson, Phys. Rev. A16 (1977) 1665. [11] R. Bonifacio and L.A. Lugiato, Phys. Rev. A18 (1978) 1129. [12] H.J. Carmichael and J.A. Hermann, Z. Phys. B28 (1980) 365. [13] J.A. Hermann, Opt. Acta 27 (1980) 159.
(23)
+ 4(1 +
155
PHYSICS LETTERS
where z~is the atomic detuning. The quantities kea = k~~(S~ç + Sk), kbb = kab(S~ Sk) are linear susceptibilities at frequency wk for levels a, b, respectively. At steady state we find 2Ix 2)~’, (7) D=N(1 ÷62)(l +62 + Ix 1~ 2I P = N(y 12(1 + i6)x~x 11/71)’ (8) X (1+62 +Jx 21x 2)--1, 11 21 where Xk = X1/2(~yj)1/4Ek are the fields appropriately scaled. In single-photon OB the mean-field limit has been defined [11] ,and examined extensively [12,13] ; we adopt essentially the same definition here. With mirror transmissivity T(0 < T< 1), we set --
g
~N(y
1I2c~
0,
T~0. (9)
11/71)
C 2gL/T-+ constant, Cbeing the cooperativity parameter and L the length of the medium, and also impose the conditions for transmitted fields (xkT) and incident fields (xkI) __
IXkTI ~ 0,
IXkl~ 0;
29 September 1980
[12]). Eqs. (9), (11), (12) and (13) can now heapplied, whereupon the coupled equations of state 2 ~ [i
=
+
— —
1 +62÷X~~l
+
[C[X76 L
+
(y~/y)”2S~(1+ 62)] ~k] 1+6 2 +x~x~
2
(14)
1, 2 (1 k), are obtained, with Yk = xkl~/T1/2,Xk = xkTI/T”2 and ~tk = Xk nNLC~ISj~.En the two-photon Raman process, the right-hand sides of eqs. (1) and (2) nmst exhibit opposing signs; this leads again to eq. (14) with, however, a negative sign preceding C. Conditions (11) and (12) apply more generally when T is no longer small, with the result k, 1
=
=
T~2{Ixk(O)12
+
R2X~ (15)
2RIxk(O)IXkcos[~k + ~ ~ where ~pk(z)are the phases of the fields Ek(z), and R = I T. It remains to evaluate xk(0) in terms of Xk --
~‘
(10) xkTI/T112
-÷
const.,
IxkII/T”2
-+
const.
The subsequent development is analogous to that specified for one-photon OB by Carmichael and Herniann [12], thus we have the boundary conditions Xk(L)
=
xkT/T’12,
and ~k and for this purpose we separate the field equations into real and imaginary parts, thus: dIxk~= —2g dz 1
--
(11)
x 26 +eSk(l +62) + x 21x 2 11 21
d~k T~2xkIei0kT=xk(0)(1 ~~T)xk(L)e2iokRe1Kk(L+L.),
where 0kT’ 0kR are the phase changes attending transmission and reflection, andL’ is the cavity path external to the active medium. We conveniently set ~k2~
—2g ~ 11 ~
~
(16a)
(16b)
~
with e = (y~/y~)1I2 and = + 2gcS~z.Eqs. (l6a) imply the spatial conservation law (Manley—Rowe relationship) for absorption:
[Kk(L +L’)+2OkR] mod 2n’
and assume
1x 2 1x 2 = 1x 2 x 2. (17) For1(z)1 the Raman 2(z)1 process the 1(0)1 conservation 2(0)1law is similar, with addition signs replacing the subtraction signs in (17). The integrals of eqs. (16a) lead to —
__
0; Xk = (1 T)T~’~k const. (13) With the time derivatives in eqs. (1), (2) set to zero, and substituting the stationary values (7) and (8) of P and D, the two field equations can be integrated over space. All quantities F(z) are replaced by ~ = L1 —
-~
X f~F(z) dz, and fields ~k can be replaced with xk(L) since the mean-field limit implies linearity (see, e.g. ref. 154
IxkHxlI 1x 2~x 2 1 2~
+ 62 +
—
—CT= 4Ix
2 Ixi(0)I2) 1(L)~ x 2(L)x1 (0) :L 6 2(a) d/l
±
ln
~
--
+ ~
(1 8a)
Volume 79A, number 2,3
PHYSICS LETTERS
with ~ = 1x1(z) 2, j3~= x1 (0)12 T 1x2(0)I 2, and i = 1 for double absorption (upper sign) and i = 2 for the Raman case (lower sign). Also eqs. (16b) produce zL
f
—
(18b)
where ~3’= Ixk(z)I 2~ The logarithmic terms, as well as the Stark terms, make the analysis very difficult; where the Stark terms and detunings may be ignored, however, the phases ~k(z) can be set to zero, with the consequence that eq. (15) becomes xk(0) = TYk + RXk. (19)
are possible. We hope to pursue the analysis further in order to ascertain the effects of the Stark terms upon the switching processes. These calculations, and also numerical results, will be reported elsewhere. Future work is also expected to deal with the more general stability condition. One of us (JAH) wishes to acknowledge financial
The conservation identities evaluated at z = L are now
support from the Science Research Council. Helpful comments were Dr. G. New, Dr. J. Elgin, Dr. P. Knight andgiven Dr. by H. Carmichael.
+
z0
d~’{6~’)/~’
As the Stark terms have been ignored here, the equation of state is 2 —2Rp1/2cos(~ ~6lnp)] l/2~ Y XT~ [p +R (24) For full transmissivity (T = 1) we have Y = Xp1/2. Although the essential features of two-photon bistable behaviour are present in eq. (24), it [2] that large effects attributable tohas the been Starkshown terms
—
~k(0) = ~
29 September 1980
eSk [1 + 62(~’)1~3’~31
—
~0~’
~ X~= (TY~+RX 2 ~ (TV 2, 1) 2 +RX2) or alternatively
(20)
References
(Y 2 —X2)[Y2 +X2(1 +R)/T] (20’) =
±(Y1 X1)[Y1 +X1(1 +R)/T] —
.
As in ref. [2] we examine the effect of varying y1 ,
with Y2 held constant. Since V1 ~X1 we see that in double absorption resonance V2 ~ X2, and in Raman resonance Y2 ~ X2, thus the switching processes are quite different for each case. With the degenerate case of double absorption, in which only one field is involved, the integration of eqs. (16a) yields (dropping all k indices and redefining kab, so that C is changed by a factor of one half) —2CT = Ix(L) 12
—
Ix(0)I 2
—
Ix(L)I —2
z=L
+
J’
~—2~2~
+
Ix(0)I —2 (21)
d13.
z0
The absence of logarithmic terms makes possible an exact analytic solution. In particular, when ~ is a constant, eq. (21) rearranges to 2p(X, 6), (22) Ix(0)I 2 = X 2p(X,6) 1 +2CTX2—(l +62)X4 +
{[1 +2CTX2 —(1 ÷62)X4]2
Eli
F.T. Arecchi (1978) 65. and A. Politi, Lett. Nuovo Cimento 23
[21G.P.
Agrawal and C. Flytzanis, Phys. Rev. Lett. 44 (1980) 1058. [3] R. Bonifacio and L.A. Lugiato, Lett. Nuovo Cimento 21 (1978) 505, 510, 517. [4] R. Bonifacio, M. Gronchi and L.A. Lugiato, Nuovo Cimento (1979). [5] R. Bonjfacio, M. Gronchi and L.A. Lugiato, Theory of optical bistability, in: Laser spectroscopy IV, Proc. 4th Conf. on Laser spectroscopy (1979), eds. Walther and Rotlie (Springer, 1979). [6] S.S. Hassan, P.D. Drummond and D.F. Walls, Opt. Commun. 27 (1978) 480. [71M. Takatsuji, Phys. Rev. Al 1(1975) 619. [81 R.G. Brewer and E.L. Hahn, Phys. Rev. All (1975) 1641. [9] N. Dutta, Phys. Lett. 69A (1978) 21. [101 L.M. Narducci, W.W. Eidson, P. Furcinitti and D.C. Eteson, Phys. Rev. A16 (1977) 1665. [11] R. Bonifacio and L.A. Lugiato, Phys. Rev. A18 (1978) 1129. [12] H.J. Carmichael and J.A. Hermann, Z. Phys. B28 (1980) 365. [13] J.A. Hermann, Opt. Acta 27 (1980) 159.
(23)
+ 4(1 +
155