- Email: [email protected]
Transient analysis of phase conjugation in semiconductor material
Volume 69, number 5,6
OPTICS COMMUNICATIONS
15 January 1989
TRANSIENT ANALYSIS OF P H A S E C O N J U G A T I O N IN S E M I C O N D U C T O R MATERIAL M.M. STALLARD and N. M A N D O K O R O Department of Electronicand ElectricalEngineering, UniversityCollegeLondon, TorringtonPlace,London WCIE 7JE, UK Received 30 May 1988
We consider conditions which favour amplified phase conjugation in GaAs material. We present a study of the transient behaviour of this effect in GaAs, not previously reported and we show amplification is favoured by gain conditions. Parameters controlling the evolution of this amplified responseare addressed.
1. Introduction
nonlinear carrier dependent susceptibility x ( N ) is given by
The integration of nonlinear devices based on degenerate and nearly degenerate four wave mixing ( F W M ) in III-V semiconductor material is attractive for many applications, in particular for optical communications [ 1,2 ] and optical computing [ 3 ]. The suitability and merit of GaAs for such applications is examined in this letter by considering the operating requirements for amplification. High efficiency FWM has been predicted in laser diodes [4 ] with milliwatt pump power. However, we present the first study of transient FWM in gain devices, and in addition we show that passive, or purely optically generated phase conjugation does not exhibit amplification in GaAs. We conclude by observing that amplified phase conjugation in a gain device may be controlled electrically, which may be a useful feature for device design.
z ( N ) = ( n c / m ) ( f l + i ) a[N(L t) - N , ] / ( 1 + l / I s ) ,
2. Theory FWM in semiconductors relies on the complex nonlinear susceptibility in which both real and imaginary components contribute. The resulting conjugate beam is also influenced by the higher order terms in addition to the third order term. The predominant nonlinear mechanism, at room temperature, in GaAs has been identified as band filling [ 7,8 ] and we consider this mechanism in this letter. The
(1) where n, c, 09 are refractive index, speed of light, and frequency respectively, a is the cross-section for stimulated processes and fl is the antiguiding factor and Is is the saturation intensity, Is =hm/atc, tc is the carrier lifetime. Nt is the cartier density corresponding to the transparency condition and N(I, t) is the intensity dependent carrier density which is given by
dN(L t ) / d t = J / e V - N ( L -
t)/tc
(I/hog) a[N(L t ) - N t ] ,
(2)
where I is the average pump intensity in the device [ 4 ], Vis the active volume, hm is the photon energy, J is the current. We are considering a laser or waveguide geometry, consequently the relevant dimensions are smaller than the diffusion length and hence we neglect effects due to diffusion in eq. (2). Combining eq. ( 1 ) and (2) with the wave equation, the amplitude of the phase conjugate wave is deduced in terms of pump (A~, A 2) and probe (A3) amplitudes (fig. 1 ). Defining the reflectivity as the ratio of the conjugate intensity to the probe intensity we can express it in terms of device parameters as follows [ 46] R=
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
~csin(HL) H cos(-~)+ ~n(tlL
2
)
'
(3)
433
Volume 69, number 5,6
OPTICS C O M M U N I C A T I O N S
AI/
K=
/ /
oH/l~
( 1 +I/I~) 2
× e x p [ - (l +I/Is)t/tc] ,
(8)
where a is the small signal absorption coefficient. Substituting eqs. (7) and (8) into eq. (3), we arrive at an expression for time resolved reflectivity.
/ -/ SEMICONDUCTOR
/
A~
(fl+i)
X [l+q+(l+I/IO(q+I/Is)t/tc--l+q]
// A3
-
15 January 1989
/ //
/
3. Results
Fig. 1. Schematic of four wave mixing geometry. Counterpropagating p u m p wavefronts of amplitude A~ and A2 respectively are incident at some arbitrary with respect to the probe wave (amplitude A3 ) on a piece of GaAs semiconductor material.
where the coupling coefficient is
x= (~o/~zc) [dN(l, t ) / d I ] l ,
(4)
and the absorption coefficient is given by
7=(ioJ/2~c){-z(N)+ [dz(N)/dl]l},
(5)
and H = [ I~:12-Re(y)2] ~/2 Under transient conditions we assume the ideal case of sharp step-like pump and probe waves, representing the transient switch-on the waves. The cartier density is given by solving eq. (2) with the initial condition N(I, t = 0 ) where I( t ) is a step-like function
N(l,t)=Np{1-exp[-(l+I/I~)t/tc]},
(6a)
where Np =Ntq( 1 + I / q l s ) / ( 1 + I / I s ) ,
(6b)
q=Jtc/eVNt .
(6c)
Thus the carrier dependent coupling coefficient and absorption are
y= [ i ( f l + i ) a / 2 ( 1 + I / I 0 2] × { ( q - 1 ) ( 2 I / I s + 1 ) - [q(2I/Is+ 1)+ (I/Is) 2
+ (q+I/Is) ( 1 +I/Is) ( I / L ) t/tc] × exp[-(l+I/l~)t/tc]}, 434
(7)
The steady state reflectivity is a function of two parameters, optical pump intensity and injection current, as illustrated in fig. 2a. The zero current case, which represents passive operation, does not exhibit an amplified reflectivity ( R > 1 ). However, for predetermined device parameters the current may be adjusted to some critical value Jc to optimise the conjugate reflectivity. The magnitude of arc is sensitive to the absorption length of the device, fig. 2a and for c~L = 1 is of the order of 10 mA. A reduction in length favours small Jc since the condition for transparency is altered, as can be seen by fig. 2b. The steady state solution (eq. (6b)) is shown for three different absorption lengths. The vertical scale (representing the number density) has been adjusted in each case for ease of presentation. A comparison of figs. 2a and 2b clearly identifies the characteristic dip in reflectivity with the transparency condition. The magnitude of the maximum achievable reflectivity is insensitive to arc, but, however, is controlled by I as can be seen in fig. 3. The maximum reflectivity is achieved when I ~ 0.1Is. The necessary condition that coupling effects dominate over absorption effects is also satisfied, fig. 5b. The transient evolution of the conjugate reflectivity displays a characteristic peak in the build up to the steady state condition, fig. 4. The influence of current on this build up is indicated by comparing the response for two different values of injection current in fig. 4a and the sensitivity of the response to the optical pump intensity is depicted in fig. 4b. The necessary condition that the coupling effects must dominate, fig. 5b to achieve amplification is again satisfied.
Volume 69, number 5,6
OPTICS C O M M U N I C A T I O N S
'°61 (a)
,1=2 1
1o3 1
4 R 10
- -
0.5
15 January 1989
--.....
/
.." /',. ..,/1
10
\
J=9.6mA
2
.," i 10 0 10
0 _
I0 - 3 -2 10
- .,~j
I
6
lO - 10
- 5
10
5
CURRENT
14
T0
(mA)
-2
1
-1
10
"x,
I#
101
WIs NP//Nt (au,)
N3 t
......
""" ,,,," . _ . " f ,," ,,"" / -' , -., / /-
,-
Fig. 3. Predicted behaviour of reflectivity as a function of optical pump intensity.
/"" / ,4-<. , .
0
10 / b 1() 2
(b) -!0
-5
0
5
10
14
~6G~
CURRENT (mA)
Fig. 2. (a) The behaviour of the phase conjugate reflectivity (R) as a function of applied current. The zero current case represents a passive device. Curves a, b and c represent intensity conditions 1~Is = 5, 1 and 0.1 respectively. The influence of altering the absorption length is indicated for the case of I/Is=O.1. (b) The steady state carrier density normalised to the threshold value N for three absorption lengths, using the same convention as in fig. 1 (a). The three intersecting lines in each case represent l/Is = 5, 1,0.1 as in (a). The intersection points correspond to the threshold value N~, N , z and N 3 (N~ is threshold of ith absorption length ).
Fig. 4. (a) Time resolved phase conjugate reflectivity illustrating the dependence on optical pump intensity. The operating conditions are: injection currents = 5 mA, absorption length = 1. Curves a, b and c correspond to 1/,I, = 5, 1 and 0.1 as before. The time is normalised with respect to the carrier lifetime. (b) Time resolved phase conjugation illustrating the dependence on injection current. Curves a and b correspond to 10 and 5 mA respectively. The time is normalised as in (a) and the absorption length is set at one.
0
I
I
I
2
(a)
NO R M A LISED TIME 10 2
10
Ib
0
~ o.1
163
1()6
L 1 NORMALISED
1
(b)
2 TIME
435
Volume 69, number 5,6
OPTICS C O M M U N I C A T I O N S
4
L~
J = lOmA
LU
Is= 0,1
0u
g g
z
D_&
__
o
(a)
[
0
1
2
NORMALI SED TIME
'/is = 0.1
z
~_ 0.5
u
c~ O
0.3 - (b) 10
I 5
0 CURRENT(mA)
5
Z 10
14
Fig. 5. (a) Time resolved coupling coefficient and absorption coefficient, current is optimised to 10 mA and I/Is is also optimised (0.1). Both coefficients are normalised with respect to the same arbitrary constant. (b) Steady state dependence of the norntalised absorption and coupling coefficient on applied current. The optical pump intensity is taken as the o p t i m u m value 0. lls and the absorption length is 1, (o~L).
4. Discussion and conclusions
Conditions favouring amplified phase conjugate reflectivity in GaAs have been identified with the conditions for gain. The maximum amplified reflectivity is achieved by appropriate choice of optical pump intensity and injection current. The magnitude of the critical I contrasts to that reported for two level atomic systems where I ~ Is. The difference arises from both the alternative nonlinear mechanism and the optical power averaging processes [4 ]. 436
15 January 1989
The predicted operating intensity favours low power operation. In the steady state, the significance of the dependence of Jc on the absorption length is clear, shorter devices require smaller injection current. In fact, the dependence of the reflectivity on both the injection current and the optical power provides additional flexibility such as electrical control, which may be advantageous in certain applications. In high speed device applications we are concerned with the speed at which steady state conditions are established. The transient response reveals that high optical pump intensities favour rapid onset of the steady state. Large injection currents have a similar effect on the transient evolution of the steady state. We can see that minimising degradation arising from transient characteristics conflicts with optimising reflectivity in the steady state. The reflectivity exhibits features which have been identified with the carrier density reaching the transparency level. Additional small contributions to the reflectivity due to alternative mechanisms, such as plasma effects [9] will raise these minimum points slightly, but will not appreciably affect the maximum points. The transient analysis exhibits a characteristic peak in the evolution to the steady state, which has previously been associated with higher order nonlinear terms [10]. In contrast to the two level case, however, this transient reflectivity is generally smaller than the steady state value. Consequently it is important to establish steady state conditions as quickly as possible. The active volume is an additional parameter which affects the behaviour of the reflectivity. We examined the implications of a change in this volume in fig. 2a by considering different absorption lengths and noted that Jc was affected. The active volume can also be altered by adjusting the crosssectional area, which results in unchanged critical current, but instead altered phase conjugate reflectivity. The cross-sectional area employed in the above analysis was fixed to be 1 ~tm 2. In conclusion, the steady state and transient behaviour of four wave mixing in GaAs material has been addressed. It was found that amplification is only possible under gain conditions. Amplification of the order of 1 0 3 - 1 0 4 is predicted for modest in-
Volume 69, number 5,6
OPTICS COMMUNICATIONS
j e c t i o n c u r r e n t s (10 m A ) a n d optical p u m p powers (0.11s). T h e t r a n s i e n t reflectivity is in general less t h a n the steady state value, a n d the onset o f the steady state m a y be controlled b y the optical p o w e r a n d the injection current. These characteristics could be used to a d v a n t a g e in optical c o m m u n i c a t i o n s a n d systems devices.
References [ 1 ] A. Yariv, IEEE J. Quantum Electron. QE-I 4 ( 1978 ) 650.
15 January 1989
[2] K. Vahala, K. Kyuma and A. Yariv, Appl. Phys. Lett. 49 (1986) 1563. [3] S.W. Koch, N. Peyghambarian and H.M. Gibbs, J. Appl. Phys. 63 (1988) RI. [4] G.P. Agrawal, Optics Lett. 12 (1987) 260. [ 5 ] Y. Silberger and I. Bar-Joseph, IEEE J. Quant. Electron. QE17 (1982) 9. [6] R.L. Abrams and R.C. Lind, Optics Lett. 20 (1978) 94. [ 7 ] D.A.B. Miller, C.T. Seaton, M.E. Prise and S.D. Smith, Phys. Rev. Lett. 47 ( 1981 ) 197. [8] B.S. Wherret and N.A. Higgins, Proc. R. Soc. Lond. A 379 (1982) 67. [9 ] R.K. Jain and M.B. Klein, Appl. Phys. Lett. 49 (1986) 1505. [ 10 ] H. Fujiwara and K. Nakagawa, J. Opt. Soc. Am. B 4 ( 1987 ) 121.
437
OPTICS COMMUNICATIONS
15 January 1989
TRANSIENT ANALYSIS OF P H A S E C O N J U G A T I O N IN S E M I C O N D U C T O R MATERIAL M.M. STALLARD and N. M A N D O K O R O Department of Electronicand ElectricalEngineering, UniversityCollegeLondon, TorringtonPlace,London WCIE 7JE, UK Received 30 May 1988
We consider conditions which favour amplified phase conjugation in GaAs material. We present a study of the transient behaviour of this effect in GaAs, not previously reported and we show amplification is favoured by gain conditions. Parameters controlling the evolution of this amplified responseare addressed.
1. Introduction
nonlinear carrier dependent susceptibility x ( N ) is given by
The integration of nonlinear devices based on degenerate and nearly degenerate four wave mixing ( F W M ) in III-V semiconductor material is attractive for many applications, in particular for optical communications [ 1,2 ] and optical computing [ 3 ]. The suitability and merit of GaAs for such applications is examined in this letter by considering the operating requirements for amplification. High efficiency FWM has been predicted in laser diodes [4 ] with milliwatt pump power. However, we present the first study of transient FWM in gain devices, and in addition we show that passive, or purely optically generated phase conjugation does not exhibit amplification in GaAs. We conclude by observing that amplified phase conjugation in a gain device may be controlled electrically, which may be a useful feature for device design.
z ( N ) = ( n c / m ) ( f l + i ) a[N(L t) - N , ] / ( 1 + l / I s ) ,
2. Theory FWM in semiconductors relies on the complex nonlinear susceptibility in which both real and imaginary components contribute. The resulting conjugate beam is also influenced by the higher order terms in addition to the third order term. The predominant nonlinear mechanism, at room temperature, in GaAs has been identified as band filling [ 7,8 ] and we consider this mechanism in this letter. The
(1) where n, c, 09 are refractive index, speed of light, and frequency respectively, a is the cross-section for stimulated processes and fl is the antiguiding factor and Is is the saturation intensity, Is =hm/atc, tc is the carrier lifetime. Nt is the cartier density corresponding to the transparency condition and N(I, t) is the intensity dependent carrier density which is given by
dN(L t ) / d t = J / e V - N ( L -
t)/tc
(I/hog) a[N(L t ) - N t ] ,
(2)
where I is the average pump intensity in the device [ 4 ], Vis the active volume, hm is the photon energy, J is the current. We are considering a laser or waveguide geometry, consequently the relevant dimensions are smaller than the diffusion length and hence we neglect effects due to diffusion in eq. (2). Combining eq. ( 1 ) and (2) with the wave equation, the amplitude of the phase conjugate wave is deduced in terms of pump (A~, A 2) and probe (A3) amplitudes (fig. 1 ). Defining the reflectivity as the ratio of the conjugate intensity to the probe intensity we can express it in terms of device parameters as follows [ 46] R=
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
~csin(HL) H cos(-~)+ ~n(tlL
2
)
'
(3)
433
Volume 69, number 5,6
OPTICS C O M M U N I C A T I O N S
AI/
K=
/ /
oH/l~
( 1 +I/I~) 2
× e x p [ - (l +I/Is)t/tc] ,
(8)
where a is the small signal absorption coefficient. Substituting eqs. (7) and (8) into eq. (3), we arrive at an expression for time resolved reflectivity.
/ -/ SEMICONDUCTOR
/
A~
(fl+i)
X [l+q+(l+I/IO(q+I/Is)t/tc--l+q]
// A3
-
15 January 1989
/ //
/
3. Results
Fig. 1. Schematic of four wave mixing geometry. Counterpropagating p u m p wavefronts of amplitude A~ and A2 respectively are incident at some arbitrary with respect to the probe wave (amplitude A3 ) on a piece of GaAs semiconductor material.
where the coupling coefficient is
x= (~o/~zc) [dN(l, t ) / d I ] l ,
(4)
and the absorption coefficient is given by
7=(ioJ/2~c){-z(N)+ [dz(N)/dl]l},
(5)
and H = [ I~:12-Re(y)2] ~/2 Under transient conditions we assume the ideal case of sharp step-like pump and probe waves, representing the transient switch-on the waves. The cartier density is given by solving eq. (2) with the initial condition N(I, t = 0 ) where I( t ) is a step-like function
N(l,t)=Np{1-exp[-(l+I/I~)t/tc]},
(6a)
where Np =Ntq( 1 + I / q l s ) / ( 1 + I / I s ) ,
(6b)
q=Jtc/eVNt .
(6c)
Thus the carrier dependent coupling coefficient and absorption are
y= [ i ( f l + i ) a / 2 ( 1 + I / I 0 2] × { ( q - 1 ) ( 2 I / I s + 1 ) - [q(2I/Is+ 1)+ (I/Is) 2
+ (q+I/Is) ( 1 +I/Is) ( I / L ) t/tc] × exp[-(l+I/l~)t/tc]}, 434
(7)
The steady state reflectivity is a function of two parameters, optical pump intensity and injection current, as illustrated in fig. 2a. The zero current case, which represents passive operation, does not exhibit an amplified reflectivity ( R > 1 ). However, for predetermined device parameters the current may be adjusted to some critical value Jc to optimise the conjugate reflectivity. The magnitude of arc is sensitive to the absorption length of the device, fig. 2a and for c~L = 1 is of the order of 10 mA. A reduction in length favours small Jc since the condition for transparency is altered, as can be seen by fig. 2b. The steady state solution (eq. (6b)) is shown for three different absorption lengths. The vertical scale (representing the number density) has been adjusted in each case for ease of presentation. A comparison of figs. 2a and 2b clearly identifies the characteristic dip in reflectivity with the transparency condition. The magnitude of the maximum achievable reflectivity is insensitive to arc, but, however, is controlled by I as can be seen in fig. 3. The maximum reflectivity is achieved when I ~ 0.1Is. The necessary condition that coupling effects dominate over absorption effects is also satisfied, fig. 5b. The transient evolution of the conjugate reflectivity displays a characteristic peak in the build up to the steady state condition, fig. 4. The influence of current on this build up is indicated by comparing the response for two different values of injection current in fig. 4a and the sensitivity of the response to the optical pump intensity is depicted in fig. 4b. The necessary condition that the coupling effects must dominate, fig. 5b to achieve amplification is again satisfied.
Volume 69, number 5,6
OPTICS C O M M U N I C A T I O N S
'°61 (a)
,1=2 1
1o3 1
4 R 10
- -
0.5
15 January 1989
--.....
/
.." /',. ..,/1
10
\
J=9.6mA
2
.," i 10 0 10
0 _
I0 - 3 -2 10
- .,~j
I
6
lO - 10
- 5
10
5
CURRENT
14
T0
(mA)
-2
1
-1
10
"x,
I#
101
WIs NP//Nt (au,)
N3 t
......
""" ,,,," . _ . " f ,," ,,"" / -' , -., / /-
,-
Fig. 3. Predicted behaviour of reflectivity as a function of optical pump intensity.
/"" / ,4-<. , .
0
10 / b 1() 2
(b) -!0
-5
0
5
10
14
~6G~
CURRENT (mA)
Fig. 2. (a) The behaviour of the phase conjugate reflectivity (R) as a function of applied current. The zero current case represents a passive device. Curves a, b and c represent intensity conditions 1~Is = 5, 1 and 0.1 respectively. The influence of altering the absorption length is indicated for the case of I/Is=O.1. (b) The steady state carrier density normalised to the threshold value N for three absorption lengths, using the same convention as in fig. 1 (a). The three intersecting lines in each case represent l/Is = 5, 1,0.1 as in (a). The intersection points correspond to the threshold value N~, N , z and N 3 (N~ is threshold of ith absorption length ).
Fig. 4. (a) Time resolved phase conjugate reflectivity illustrating the dependence on optical pump intensity. The operating conditions are: injection currents = 5 mA, absorption length = 1. Curves a, b and c correspond to 1/,I, = 5, 1 and 0.1 as before. The time is normalised with respect to the carrier lifetime. (b) Time resolved phase conjugation illustrating the dependence on injection current. Curves a and b correspond to 10 and 5 mA respectively. The time is normalised as in (a) and the absorption length is set at one.
0
I
I
I
2
(a)
NO R M A LISED TIME 10 2
10
Ib
0
~ o.1
163
1()6
L 1 NORMALISED
1
(b)
2 TIME
435
Volume 69, number 5,6
OPTICS C O M M U N I C A T I O N S
4
L~
J = lOmA
LU
Is= 0,1
0u
g g
z
D_&
__
o
(a)
[
0
1
2
NORMALI SED TIME
'/is = 0.1
z
~_ 0.5
u
c~ O
0.3 - (b) 10
I 5
0 CURRENT(mA)
5
Z 10
14
Fig. 5. (a) Time resolved coupling coefficient and absorption coefficient, current is optimised to 10 mA and I/Is is also optimised (0.1). Both coefficients are normalised with respect to the same arbitrary constant. (b) Steady state dependence of the norntalised absorption and coupling coefficient on applied current. The optical pump intensity is taken as the o p t i m u m value 0. lls and the absorption length is 1, (o~L).
4. Discussion and conclusions
Conditions favouring amplified phase conjugate reflectivity in GaAs have been identified with the conditions for gain. The maximum amplified reflectivity is achieved by appropriate choice of optical pump intensity and injection current. The magnitude of the critical I contrasts to that reported for two level atomic systems where I ~ Is. The difference arises from both the alternative nonlinear mechanism and the optical power averaging processes [4 ]. 436
15 January 1989
The predicted operating intensity favours low power operation. In the steady state, the significance of the dependence of Jc on the absorption length is clear, shorter devices require smaller injection current. In fact, the dependence of the reflectivity on both the injection current and the optical power provides additional flexibility such as electrical control, which may be advantageous in certain applications. In high speed device applications we are concerned with the speed at which steady state conditions are established. The transient response reveals that high optical pump intensities favour rapid onset of the steady state. Large injection currents have a similar effect on the transient evolution of the steady state. We can see that minimising degradation arising from transient characteristics conflicts with optimising reflectivity in the steady state. The reflectivity exhibits features which have been identified with the carrier density reaching the transparency level. Additional small contributions to the reflectivity due to alternative mechanisms, such as plasma effects [9] will raise these minimum points slightly, but will not appreciably affect the maximum points. The transient analysis exhibits a characteristic peak in the evolution to the steady state, which has previously been associated with higher order nonlinear terms [10]. In contrast to the two level case, however, this transient reflectivity is generally smaller than the steady state value. Consequently it is important to establish steady state conditions as quickly as possible. The active volume is an additional parameter which affects the behaviour of the reflectivity. We examined the implications of a change in this volume in fig. 2a by considering different absorption lengths and noted that Jc was affected. The active volume can also be altered by adjusting the crosssectional area, which results in unchanged critical current, but instead altered phase conjugate reflectivity. The cross-sectional area employed in the above analysis was fixed to be 1 ~tm 2. In conclusion, the steady state and transient behaviour of four wave mixing in GaAs material has been addressed. It was found that amplification is only possible under gain conditions. Amplification of the order of 1 0 3 - 1 0 4 is predicted for modest in-
Volume 69, number 5,6
OPTICS COMMUNICATIONS
j e c t i o n c u r r e n t s (10 m A ) a n d optical p u m p powers (0.11s). T h e t r a n s i e n t reflectivity is in general less t h a n the steady state value, a n d the onset o f the steady state m a y be controlled b y the optical p o w e r a n d the injection current. These characteristics could be used to a d v a n t a g e in optical c o m m u n i c a t i o n s a n d systems devices.
References [ 1 ] A. Yariv, IEEE J. Quantum Electron. QE-I 4 ( 1978 ) 650.
15 January 1989
[2] K. Vahala, K. Kyuma and A. Yariv, Appl. Phys. Lett. 49 (1986) 1563. [3] S.W. Koch, N. Peyghambarian and H.M. Gibbs, J. Appl. Phys. 63 (1988) RI. [4] G.P. Agrawal, Optics Lett. 12 (1987) 260. [ 5 ] Y. Silberger and I. Bar-Joseph, IEEE J. Quant. Electron. QE17 (1982) 9. [6] R.L. Abrams and R.C. Lind, Optics Lett. 20 (1978) 94. [ 7 ] D.A.B. Miller, C.T. Seaton, M.E. Prise and S.D. Smith, Phys. Rev. Lett. 47 ( 1981 ) 197. [8] B.S. Wherret and N.A. Higgins, Proc. R. Soc. Lond. A 379 (1982) 67. [9 ] R.K. Jain and M.B. Klein, Appl. Phys. Lett. 49 (1986) 1505. [ 10 ] H. Fujiwara and K. Nakagawa, J. Opt. Soc. Am. B 4 ( 1987 ) 121.
437