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Electronic-vibrational coupling effects in the single mode laser
15 April 1996 OPTICS
COMMUNICATIONS ELSlYlVIER
Optics Communications
125 ( 1996) 369-376
Full length article
Electronic-vibrational
coupling effects in the single mode laser Oscar G. CalderBn, Isabel Gonzalo
DeIJartume~?o de Ophx, Fucuitad de Ciencius Fisicas. U~i~~er.~idad Complufense de ~udrjd, &dad Universitaria s/n. 28040Madrid, Spain Received
17 August
1995; revised version received 20 October
1995; accepted
7 December
1995
Abstract The effect of the electronic-vibrational coupling in the two-level active centers of a single-mode laser is analyzed by means of the semiclassical Maxwell-Bloch equations. it is found that the region of laser oscillation experiences significant modi~cations at the time that shifts to a lower frequency. A stationary solution, modification of the well-known continuous wave. is found.
1. Introduction In the electronic excitation of molecules or crystals by radiation, vibrational couplings may modify significantly the behaviour of the system. In this case the Maxwell-Bloch equations are further coupled by a nonlinear term which depends on the electronvibration coupling. Nonlinear optical response originating in excitonphonon transitions has been pointed out [l-5]. In particular, phonon effects on optical bistability were shown to lead to a new type of optical bistability [471. Here we study the modifications in a single mode laser system arising from the coupling between the electronic system and vibrational normal modes of the nuclei in the active medium. The treatment we follow applies to a medium composed either of molecules, or of atoms inserted as impurity centers in a crystal. Two electronic levels connected by one photon transitions arc considered. The rotating wave and the slowly varying envelope approximations are assumed in our semiclassical treatment. The new terms appearing in the
OO30-JO I 8 /96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved P/I SOO30-4018(96)00013-2
Maxwell-Bloch equations lead to a shift of the region of laser oscillation to a lower frequency, in addition to other modifications. It is found that, for a given detuning and particular conditions, there is a pump value above which laser emission disappears. Also, a modification of the well-known continuous wave (CW) stationary solution of the Maxwell-Bloch equations is found.
2. The model Our system consists of N identical molecules per unit volume inserted in a cavity. The length of the cavity is tuned to a single longitudinal mode of frequency w. Each molecule is modeled by two electronic levels only, the ground state ( l} and the excited state (2) with respective energies El and E2 in theoptical range, con: netted by one photon transitions so that ftw z E2 - El. We shall consider that the active electrons of the molecules interact with the vibrational normal modes of the nuclei {phonons if the molecule is a big polymer or we deal with a crystal) by means of a simplified model.
?70
O.G. Cafderdn, 1. Gnnzulo/Opfics
The Hanliltonian
of the efectronic
system is
H :: H” f H”“’ $- He-“,
(1)
Ho is the free electronic Hamiltonian whose eigenstales are 1I) and 12). The electronic wave function is then I(,,) = C, ( tle-=‘lk
1I} + C*(t)e-‘EZclE
j2j.
(21
The electric dipole interaction with the radiation is given by ffeer = pE, with J.LE ex, where e is the absolute value of the charge of the electron, and E the radiation electric field, linearly polarized in the x direction, the same direction as the transition and permanent dipoles. He-” is rhe electron-vibration Hamiltonian, where we assume a linear coupling between the excited electronic state 12) and only one normal mode ( that interacting stronger). To analyze better this term it is first written, following [ 8,6,7,9 1, as H”‘” = fro, (u*b + ilb’)
12)(2/ ,
(4)
Taking the new equilibrium coordinates as the new origin of reference, the quantum vibrational operator for the mode reads 7; = 0 + IC$U
,
whereupon the averaged electron-vibration can he written as
12.5 (1996) 369-376
where f z w,.(u~~ has frequency dimension and 1 + d = 2iC#, with d E i&i2 - jCr/* being the population inversion per molecule. The effective electronvibration Hamiltonian acting only on the electronic state I@) is then +d)hf
q-f” = -(I
12}(2[.
(7)
To obtain the evolution equations of the system we follow a well-known procedure 1101. From the Schrodinger equation for the function (2). the time evolution equations for Ct and C2 are
where p”ii z (ilexlj) and wt2 z (Ez - El )/FL The microscopic polarization is
(31
where 0, hi are the vibrational quanta annihilation and creation operators, w,~is the vibrational frequency, W, << w, and u is dependent on the interaction strength and related to the relative displacement of the equiIibriutn positions of the nuclei after the molecule is excited to the level 12). As the life time of the electronic excited state is much longer than the evolution time of the nuclei to the new equilibrium positions, we can evaluate the quantum mechanical average of iYe-” factoring the electronic and nuclei variables, that is, IC212 can be considered to be constant while the nuclei evolve to the new equilibrium positions corresponding to the electronic excited state: (Ht’-~J&&. = tuLl.s~c~/~(u*b + z&+)(%*
Communications
(51 interaction
= (p” +p-1
- #u,,/C,l”-
fiz2jC#,
(10)
where p’ E -1u_t2C;C2e-~~i?’ and p- = pt.*. Taking the time derivative of pf and of the population inversion per mo?~ule, d, using the expressions (8), (91, multiplying by N to introduce the macroscopic variables P = Np, D = Nd, and including phenomenologically the relaxation terms for the polarization and population inversion, we obtain
+i dD dt
c I>
=rjl(D,
i +;
fP’-
- D) - $p
ijl*.1#ED tl , - P--),
(11) c 121
where DO is the population inversion induced by the pumping, ~1’ is the depolarization time of the radiation induced dipoles, and ~0’ is the lifetime of the population inversion. The electric radiation field into the cavity, without transversal spatial effects, is given by the Maxwell wave equation 3’P I 8”E aE t12E - M”“z - -3 &2 =ruo-$a22 (I.S.units),
f 13)
O.G. Calderh,
I. Gonzalo/Optics
where gc stands for the effective cavity losses and m is the vacuum magnetic permeability. The field is assumed to be of the form, E = E,f (t) exp( ikz - iwt) + C.C. Establishing the one-photon resonance condition (w x WI*), using the rotating wave and the slowly varying amplitude approximations, Eqs. (1 l)-( 13) transform into
apo+ --(yl+iw,*-I’w)Po++ifPof
1,;
at
(
--ilp~21*E~D/h, dD = yII(Do - D) - ;(E;p; dt dE,t 2~ = -,uoc*u,E~ where P: is To obtain we define r the following
> (14)
- E,+pJ,
+ i~~c*wP~,
(15) (16)
polarization amplitude. dimensionless and simplified equations, = yet, LY= No~,ur2~*/(2&iy~) and variable changes
Communications 125 (19%)
369-376
371
strength f, while ac+D,s,depending on the population inversion, acts as a dynamic detuning.
3. Analysis of the stationary
solutions
Eqs. (19)-(21) have two stationary solutions, the trivial one E,v= 0, ev = 0, D,y= r (no laser emission), and another one E, = P.,, IE,1* = r - I, D,s = I when au=S. First, let analyze the stability of the trivial stationary state E,y = 0, P,s = 0, D,v = r. The linearized system around this solution is
a(=,) ~
=u[(@‘,) - (SE,)],
a(@,) ~
=--(I
I_%
a7
+iS)(SP,)
(22)
+iaar(~3P,~)
+r (S.%),
(23)
which yields the three eigenvalues (25)
A3 =-y,
(17)
A*=-$[1
+a+i(S-am)] -a+i(&am)]*+4i7r.
*iJ[l NU,
D= -D.,, 2eaccY
NUC Do=-1. 2Q$X_Y’
where P,, E,, D,v, r, are the new dimensionless magnitudes. In terms of these new variables, the Eqs. ( 14)( 16) take finally the form
The trivial solution is linearly unstable if one of the eigenvalues has a positive real part. The expression under the square root is a complex number which will be denoted by (x + i[)2, with x > 0, namely, [l -~+i(6-am)]*+4~r.
(x+i&)2= dE, _ =dP., - E,), $r JP, = - ( I + ia) P., + iac+D,yP,y + D,yE,y, ar dD,Y -=-~[D,~-r+~(~;q,+~,~q;)j, al-
(19) (20) (21)
where v = v~/(~EoYI), 6 = (or2 - w - f)/rl = A - f/71, with A = (0112 - w)/yl, a = f/(m), and the normalized pump r z Y = YIlh ~EOC(YDO/(Ng,). This system of equations is similar to that of Complex Lorenz Equations (CLE), except for the non-linear term iac+D,yPsand the term ifP.,/yl, in the polarization equation. f/r1 acts as a detuning originated by the vibrational coupling
(26)
(27)
It can be seen that the trivial solution is stable if and only if -( 1 + (T) f x < 0, i.e., ( I + (T)* > x2. Equating the real and imaginary parts in (27) we find that x satisfies F(~*)~~~-[(1-‘~)~-(8--am)~+4m]x~ -(I
- ~)~(a - am-)2 = 0.
(28)
The expression in the first member of Eq. (28) can be interpreted as a function F of a variable, F( x2 ), which has a real and positive zero for x2 = x2. On the other hand, it can be seen that the parabola F ( x2) has only a single real and positive zero, hence F( x2 ) > 0 for x2 > x2. In particular, if ( 1 + (T)~ > x2 (the
772
O.G. Calderh,
1. GonzalofOptics
Communicntkms
125
(I 996) 369-376
Axwc cmdi tion) , the function F takes a positive value for .$ = i I i CF)~. Hence, when x2 is replaced by i I - ~7)’ in Eq. (28) we have F(( I +a)‘) =4cr[~l$_cr)~+(~-~aar)2-r(t+lrr)21 > 0.
(29)
Thus the stability condition [ I -i- o-j2 > ,y2 is equivalent to the condition (29t, which can be rearranged to read
(35) is obtained. Then, for an electron-vibration interaction strength high enough, or for concentrations lower than a threshold one, Nthr, there is no laser emission for any value of the pump. (IS) The parabolic function in f 30) has two real zeros for a given value of the detuning - two pump boundaries - r+ and r”_, r*=(1/2&2){2Sacr+
fl>flV-
(1 +g)’
[2Satr+(l+(T)*]y$(1-t.a)*+S2 rt[(26ff~+((1+0+)~
> 0.
(30)
If the new parameter a, which depends on the vibrational coupling, is zero, condition (30) gives the wellknown first laser threshold f normalized pump ro ), A2 r <: I + (, +
c.f=O).
s ro.
a)2
(31)
In our case the condition (30) is a quadratic function of Y, which leads to the following CUSS: f A j The condition (30) is always satisfied for any value of the normalized pump Y. It means that the parabolic function in (30) has only complex zeros. For this to occur the condition for the coefficients of the parabola is > 0, which gives a condition L1:> cl~, S
(321
for the new parameter a
s+J&t(l+(T)* 2cT
a
(33)
The other zero of the function in the 1.h.s. of (32), a_, is negative and has no physical meaning. Then, under the condition ( 33) the trivial solution is always stable and laser emission cannot be obtained however much the pump be increased. Introducing a zz f/ (ecu), with cz in Nwl,u~,j*/(2~0cfiY_~), a condition for the vibrational coupI& strength f,
.f >
s+ J&iCN
(1 3
&cr
or for the concentration
+g_)* 1
N of the active centers
(341
-4(fl
+a)*es’)O”@Zj~~Z].
(36)
The condition (30) is then satisfied (the trivial solution is stable) only if r < r- or r > P+. If p- < r < r+ the trivial solution is unstable. In this case there is at least one oscillating solution for the field amplitude (although constant intensity), being similar to the solution of the CLE but with different period. For values r > T+, it was found numerically that other stable solutions in addition to the trivial one may occur; the trivial solution, the oscillating one. or both solutions coexisting, may take place depending on the initial condition. Concerning the stationary solution E,y= P,, lE,yI”= I”--I, D,y= 1, (which occurs only if ncT = 6) the stahility analysis leads to a linearized system of five equations (for E,, E,‘, P,,. P,* and D,) . The corresponding eigenvafue problem can he reduced by means of a straightforward algebraical manipulation to solve the cubic equation h”$(l+cr+y)A”t-y(l+cr+IE.~I’)/\ +2gyiE,.1* = a. The stability condition (I Ca+yfy(l
(37) is then
-W-i+/‘)
-2oy!Gj
>O, (38)
which is equivalent
to the two conditions
O.G.
Culderdn.
i.
Gonzalo/Optics
Communicuiions 125 (1996)
X9-376
24 ;23 z UI - 022
b
Fig.
I. ( a)-(d):
Decimal logarithm of the threshold molecular density versus A z (co12 - w)/y~, for different values of the vibrational f = 2 x IO I2, 6 x IO’*, 1013, 2 x IO”. (e)-(h): For each one of such values of f, the corresponding normalized
coupling parameter pumps. Y-
(thin solid line), Y+ {dashed line of long dash), the upper bound rUP (thick solid line), and rn (f
dash ). versus d. for molecular density N = IO2i m-‘.
= 0) (dashed line of short
The boundary between r.- and r+ sections is marked by a small horizontal segment.
where the solution lE,s12= r - 1 has been used. The first condition is a bad cavity condition; the second one mews that the pump must no exceed a given value.
4. Numerical
examples
Let us consider molecular data representative of some substituted aromatic molecules (for example), which are frequently used in lasers and nonhnear optics. Let we choose wt2 = lO’$ s-t (in the optical range) and the transition dipole moment of the order of the Debye ~~2 = 1 D = 3.3695 x lo-“O C m. We
,574
take y;’
O.C. Calderrin. I. C;onzulo/Optics C~lmrn~nicut~o~.~ 125 (1996) 369-376
= I W9 s, and a shorter value (as usual)
for the dephasing or decoherence time 7;’ = lo-” s. We also take N = l023 molecules rnw3, the length of the cavity L = 0.01 m, and the reflectivity of the mirrors R, = I, R2 = 0.3. The flight time of the photon into the cavity [lo], t,, then verifies C,/EO 2 tC:’ E -c In &f&/L = 1.806 x lOto s-l. Concerning the parameter f E us/u/*, the value of the vibrational frequency w, is between - 10” s-t (some bending modes in large plane molecules) and - lOI s-’ . The dimensionless parameter Iu/*, which invotves the nuclei displacement relative to their old equilibrium positions and the strength of the coupling, may take values from 1 to 20 approximately [ 111. We then take f of the order of lOI s-t or lOi s-l. The variables A z (~12 - u>/~J_ and N can be easily controlled experimentally. We then represent the threshold concentration N&, given by (35), versus A for different values of f (Fig. 1). For N = 102” me3 we can see in the graphics the detuning range where N < Nthr (case A), and N > Nthr (case B). The curve for N&r shifts towards positive values of A as f grows. The behaviour of the normalized pump ro, r+, and r_, as functions of A is also shown in Fig. 1 for each one of the values of f considered above. The wellknown laser threshold for f = 0, ro, is a parabola centered on A = 0. It is shown in the graphics for the sake of comparison when f #O. The fact that the population inversion per molecule induced by the pumping, da zz Da/N, cannot be physically greater than unity, leads to an upper bound for the pump given by rup E 2ctrt,, which is shown in the graphics. Values of I > ruP have no physical meaning. For low enough values of f, the curve r_ versus A approaches to that of ro, as expected, whereas r.+ > rup has no physical meaning. However, for greater values of f’, the values of P+ can be reached physically, and they are lower as f grows, at the time that the curves of T_+and Y_ shift towards positive values of A and slope to the right (Fig. 1e-h). That is, the eiectronicvibrational coupling shifts the region of laser oscillation to a lower frequency (higher value of A). The fact that the I+ section slops to the right allows that laser action be bounded by two pump values, r_ and r,_, at a fixed detuning (case B), so that for r > r+. laser action may disappears. Lasing occurs in the region bounded by r-, r+ and rup.
23.5
22.5
Fig. 2. Normalized pumps r_ (solid line), r+ (dashed line), bound TV,,(dotted line), versus decimal logarithm of
upper
molecular density N, for f = 2 x 10’” s-l
(Fig.
ih),
the same value of the rest of parameters as in Fig.
and the
3 = 30, and
I~
It is to be noted that for a given value of r, the population inversion per molecule induced by the pumping is do = r/rup. The dependence of the normalized pumps I_, J-+ and rupr on the concentration is shown in Fig. 2, for f = 2 x IO’” and A = 30. To calculate numerically the stationary solutions of the field in cases A and B, we choose f = 2 x IO’” (Fig. I h). For the data assumed we obtain c = 0.009 and y = 0.001. As an example of case A we take a detuning A = 5 (see Fig. Ih); then 6 = -15 and CI = 33.057. For any value of r < I^,~, the trivial solution (no laser emission) is stabIe. This has been the result using P = 10,15,20 < rg = 25.55 (no laser action for f = 0), as well as using r = 30,40,50 > r0 (laser action for f=O). As an example of case B we take A = 25 (see Fig. 1h); then 6 = 5 and a = 33.735, obtaining r- = 7.785, < r+ = 36.013 < rup = 65.654. Using r = 2,4,5,6 r_-, the solution found is also the trivial one for several initial conditions. For r_ < r < r+. (r = 10,20,30) the solution found is the oscillating one in the field amplitude. An estimation of the power, $~r~jE~j*lO-'~ MW cmv2, where EJ is related to E.y by ( 17), gives, for r = 30 ( /E,s/2 = 22.5), a value of 2.93 MW cmP2. Using r = 40,50,60 > r+, the trivial solution or (and) the oscillating one are found depending on the initial condition, contrary to the preceding cases. it is noticeable that for the same initial condition and changing r, we find the oscillating solution for Yclose
5. Conclusions
Fig. 3. The same as in Fig. 1for f = 2x lOJ3s-’ but now N = lo*’ m-‘, and cavity conditions leading to tc’ = 2.3 x lOI s-‘.
to r+, and the trivial one for greater values of r. This
would mean that the basin of the atractor for the trivial solution grows as r increases. Finally, we propose an example where the condition U(T = (0~12- w - f) /rl holds, which means that the total detuning including the dynamic one is zero, i.e., a new physical tuned situation is achieved. It corresponds to A = A, FZ c + f/y~_. In this case, the stationary solution E, = P,,, l&l2 = r- 1, D, = 1 is stable if the relations (39) are satisfied. For such conditions be satisfied with the same values of the molecular parameters, we change both the cavity and the molecular density. For the case f = 2 x IO’” s-’ , we take now L = 6x10-‘m,Rt =Rz=O.Ol (t,’ =2.303x101*s-I), and N = 10zs molecules me3 (see Fig. 3). It leads to P = I.151 > 1.001 = 1 + (q/y~), and rf = 31.808. The stationary solution occurs for A, = 20.387 (then 6 = 0.387, a = 0.336). As expected, this solution has been also obtained numerically using several values of Y < rf. The estimated power for r = 28 ( IE,y12= r - 1 = 27), e.g., is 3.514 MW cmF2.
Electronic-vibrationa coupling in the two-level active centers of a single-mode laser has been analyzed assuming a linear coupling between the excited electronic state and only one normal vibrational mode. This coupling leads to two new terms (one of them nonlinear) in the semiclassical Maxwell-Bloch equations. Such terms are proportional to the coupling strength and act as detuning terms. It can then be interpreted that the upper electronic level experiences a slight energetic shift proportional to the coupling strength, one of the contributions to this shift being also dependent on the population inversion. The result is that the electronic-vibrational coupling shifts the region of laser oscillation to a lower frequency at the time that the boundary of the region changes its slope. In addition to the trivial stationary solution (no laser emission) lZs = 0, P,7 = 0, D,s = r, the stationary solution of constant field amplitude, E, = Ps, /E,,1’= r- 1, D,y = 1, is found when the frequency of the field, w, is detuned with respect to the two initial electronic levels, 012, so that it compensates the energetic shift mentioned above and a tuned situation is achieved (ag = 8). Such a solution is just a modification of the well-known CW stationary solution for ,f = 0. In the stability analysis of the trivial solution, it is found that for values of the parameter n = f/( ecu) greater than the value given by (33) the trivial solution is always stable. Then laser emission cannot take place however much the pump be increased. This situation occurs, for example, in the case of high electronicvibrational coupling, or when the concentration of the active centers is lower than a critical vaiue Nthr given by (35). When the parameter a is lower than the value given by (33), i.e., the concentration is higher than N{hr, two boundary pumps, I_ and r,., given by Eq. (36). appear. It must be noted that r- tends to the well-known first laser threshold, r-0, as the electronic-vibrational coupling tends to zero. For the normalized pump values, r, lower than rl, the trivial solution is stable. For r_ < r < r+ the trivial solution is unstable and there is at least one stable oscillating field amplitude solution, similar to that of the CLE although with different period. For values of r higher than r+ the laser emission may disappear; the trivial solution is stable again but
i-6
O.G. Culderrin. I. Gonza6~/U~rics C~~muniru/if~ns 125 (1996) 369-376
other stable solutions may also take place. It is found that the trivial solution, the oscillating one, or both coexisting, can be obtained depending on the initial conditions. It could be understood that for high enough pump values (r > I-+), the global detuning, which depends on the population inversion, may be too high and the system be then too far from resonance. Further understanding on the behaviour of the laser system for values of r > r+ is the aim of a future work. Concerning the stationary solution of constant field anlplitud~, its stability requires a bad cavity condition and pump values lower than that given by (39). Finally we remark that similar results could also be found in other formally analogous situations where the excited state is coupled to other system as, for example, in the case of excimer formation.
References I I 1 C. Cojan, G. Agrawal and C. Flytzanis, Phys. Rev. B I5 (1997) 121 BP
909.
Antoniuk, Solid State Commun. 33 ( 1980)
131G.J.Blanchard,
71.
15 1 X.S. Li. D.L. Lin and T. George. Phys. Rev. B 41 t 19901 3280.
[ 6 1 1.Gonzalo,
M. A. Ant&
Electron. 28 (1992) 171 M.A. [S] BP
and J.L. Escudero. IEEE J. Quantum
765.
Ant&t and 1. Gonzalo, IEEE J. Quantum Electron. 31
(1995)
1088.
Antoniuk, Sov. Phys. JETP 53 ( 1981)
1159.
19 1 D. Pines, Elementary Excitations in Solids (Benjamin,
New
York. 1964).
1101H.
Haken,
Light
Vol. 2: Laser Light
Holland, Amsterdam,
[ 111 D.B.
Dynamics
f North-
1985).
Fitchen, Zero-phonon transitions, in: Physics of Color
1968).
We are very grateful to J. M. Guerra for his helpful advice.
873.
887.
141 N. Ba and N.T. Dan, Phys. Len. A 136 (1989)
Centers,
Acknowledgement
Phys. Rev. Lett. 63 ( 1989)
ed. W.B.
Fowler
(Academic
Press. New
York,
COMMUNICATIONS ELSlYlVIER
Optics Communications
125 ( 1996) 369-376
Full length article
Electronic-vibrational
coupling effects in the single mode laser Oscar G. CalderBn, Isabel Gonzalo
DeIJartume~?o de Ophx, Fucuitad de Ciencius Fisicas. U~i~~er.~idad Complufense de ~udrjd, &dad Universitaria s/n. 28040Madrid, Spain Received
17 August
1995; revised version received 20 October
1995; accepted
7 December
1995
Abstract The effect of the electronic-vibrational coupling in the two-level active centers of a single-mode laser is analyzed by means of the semiclassical Maxwell-Bloch equations. it is found that the region of laser oscillation experiences significant modi~cations at the time that shifts to a lower frequency. A stationary solution, modification of the well-known continuous wave. is found.
1. Introduction In the electronic excitation of molecules or crystals by radiation, vibrational couplings may modify significantly the behaviour of the system. In this case the Maxwell-Bloch equations are further coupled by a nonlinear term which depends on the electronvibration coupling. Nonlinear optical response originating in excitonphonon transitions has been pointed out [l-5]. In particular, phonon effects on optical bistability were shown to lead to a new type of optical bistability [471. Here we study the modifications in a single mode laser system arising from the coupling between the electronic system and vibrational normal modes of the nuclei in the active medium. The treatment we follow applies to a medium composed either of molecules, or of atoms inserted as impurity centers in a crystal. Two electronic levels connected by one photon transitions arc considered. The rotating wave and the slowly varying envelope approximations are assumed in our semiclassical treatment. The new terms appearing in the
OO30-JO I 8 /96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved P/I SOO30-4018(96)00013-2
Maxwell-Bloch equations lead to a shift of the region of laser oscillation to a lower frequency, in addition to other modifications. It is found that, for a given detuning and particular conditions, there is a pump value above which laser emission disappears. Also, a modification of the well-known continuous wave (CW) stationary solution of the Maxwell-Bloch equations is found.
2. The model Our system consists of N identical molecules per unit volume inserted in a cavity. The length of the cavity is tuned to a single longitudinal mode of frequency w. Each molecule is modeled by two electronic levels only, the ground state ( l} and the excited state (2) with respective energies El and E2 in theoptical range, con: netted by one photon transitions so that ftw z E2 - El. We shall consider that the active electrons of the molecules interact with the vibrational normal modes of the nuclei {phonons if the molecule is a big polymer or we deal with a crystal) by means of a simplified model.
?70
O.G. Cafderdn, 1. Gnnzulo/Opfics
The Hanliltonian
of the efectronic
system is
H :: H” f H”“’ $- He-“,
(1)
Ho is the free electronic Hamiltonian whose eigenstales are 1I) and 12). The electronic wave function is then I(,,) = C, ( tle-=‘lk
1I} + C*(t)e-‘EZclE
j2j.
(21
The electric dipole interaction with the radiation is given by ffeer = pE, with J.LE ex, where e is the absolute value of the charge of the electron, and E the radiation electric field, linearly polarized in the x direction, the same direction as the transition and permanent dipoles. He-” is rhe electron-vibration Hamiltonian, where we assume a linear coupling between the excited electronic state 12) and only one normal mode ( that interacting stronger). To analyze better this term it is first written, following [ 8,6,7,9 1, as H”‘” = fro, (u*b + ilb’)
12)(2/ ,
(4)
Taking the new equilibrium coordinates as the new origin of reference, the quantum vibrational operator for the mode reads 7; = 0 + IC$U
,
whereupon the averaged electron-vibration can he written as
12.5 (1996) 369-376
where f z w,.(u~~ has frequency dimension and 1 + d = 2iC#, with d E i&i2 - jCr/* being the population inversion per molecule. The effective electronvibration Hamiltonian acting only on the electronic state I@) is then +d)hf
q-f” = -(I
12}(2[.
(7)
To obtain the evolution equations of the system we follow a well-known procedure 1101. From the Schrodinger equation for the function (2). the time evolution equations for Ct and C2 are
where p”ii z (ilexlj) and wt2 z (Ez - El )/FL The microscopic polarization is
(31
where 0, hi are the vibrational quanta annihilation and creation operators, w,~is the vibrational frequency, W, << w, and u is dependent on the interaction strength and related to the relative displacement of the equiIibriutn positions of the nuclei after the molecule is excited to the level 12). As the life time of the electronic excited state is much longer than the evolution time of the nuclei to the new equilibrium positions, we can evaluate the quantum mechanical average of iYe-” factoring the electronic and nuclei variables, that is, IC212 can be considered to be constant while the nuclei evolve to the new equilibrium positions corresponding to the electronic excited state: (Ht’-~J&&. = tuLl.s~c~/~(u*b + z&+)(%*
Communications
(51 interaction
= (p” +p-1
- #u,,/C,l”-
fiz2jC#,
(10)
where p’ E -1u_t2C;C2e-~~i?’ and p- = pt.*. Taking the time derivative of pf and of the population inversion per mo?~ule, d, using the expressions (8), (91, multiplying by N to introduce the macroscopic variables P = Np, D = Nd, and including phenomenologically the relaxation terms for the polarization and population inversion, we obtain
+i dD dt
c I>
=rjl(D,
i +;
fP’-
- D) - $p
ijl*.1#ED tl , - P--),
(11) c 121
where DO is the population inversion induced by the pumping, ~1’ is the depolarization time of the radiation induced dipoles, and ~0’ is the lifetime of the population inversion. The electric radiation field into the cavity, without transversal spatial effects, is given by the Maxwell wave equation 3’P I 8”E aE t12E - M”“z - -3 &2 =ruo-$a22 (I.S.units),
f 13)
O.G. Calderh,
I. Gonzalo/Optics
where gc stands for the effective cavity losses and m is the vacuum magnetic permeability. The field is assumed to be of the form, E = E,f (t) exp( ikz - iwt) + C.C. Establishing the one-photon resonance condition (w x WI*), using the rotating wave and the slowly varying amplitude approximations, Eqs. (1 l)-( 13) transform into
apo+ --(yl+iw,*-I’w)Po++ifPof
1,;
at
(
--ilp~21*E~D/h, dD = yII(Do - D) - ;(E;p; dt dE,t 2~ = -,uoc*u,E~ where P: is To obtain we define r the following
> (14)
- E,+pJ,
+ i~~c*wP~,
(15) (16)
polarization amplitude. dimensionless and simplified equations, = yet, LY= No~,ur2~*/(2&iy~) and variable changes
Communications 125 (19%)
369-376
371
strength f, while ac+D,s,depending on the population inversion, acts as a dynamic detuning.
3. Analysis of the stationary
solutions
Eqs. (19)-(21) have two stationary solutions, the trivial one E,v= 0, ev = 0, D,y= r (no laser emission), and another one E, = P.,, IE,1* = r - I, D,s = I when au=S. First, let analyze the stability of the trivial stationary state E,y = 0, P,s = 0, D,v = r. The linearized system around this solution is
a(=,) ~
=u[(@‘,) - (SE,)],
a(@,) ~
=--(I
I_%
a7
+iS)(SP,)
(22)
+iaar(~3P,~)
+r (S.%),
(23)
which yields the three eigenvalues (25)
A3 =-y,
(17)
A*=-$[1
+a+i(S-am)] -a+i(&am)]*+4i7r.
*iJ[l NU,
D= -D.,, 2eaccY
NUC Do=-1. 2Q$X_Y’
where P,, E,, D,v, r, are the new dimensionless magnitudes. In terms of these new variables, the Eqs. ( 14)( 16) take finally the form
The trivial solution is linearly unstable if one of the eigenvalues has a positive real part. The expression under the square root is a complex number which will be denoted by (x + i[)2, with x > 0, namely, [l -~+i(6-am)]*+4~r.
(x+i&)2= dE, _ =dP., - E,), $r JP, = - ( I + ia) P., + iac+D,yP,y + D,yE,y, ar dD,Y -=-~[D,~-r+~(~;q,+~,~q;)j, al-
(19) (20) (21)
where v = v~/(~EoYI), 6 = (or2 - w - f)/rl = A - f/71, with A = (0112 - w)/yl, a = f/(m), and the normalized pump r z Y = YIlh ~EOC(YDO/(Ng,). This system of equations is similar to that of Complex Lorenz Equations (CLE), except for the non-linear term iac+D,yPsand the term ifP.,/yl, in the polarization equation. f/r1 acts as a detuning originated by the vibrational coupling
(26)
(27)
It can be seen that the trivial solution is stable if and only if -( 1 + (T) f x < 0, i.e., ( I + (T)* > x2. Equating the real and imaginary parts in (27) we find that x satisfies F(~*)~~~-[(1-‘~)~-(8--am)~+4m]x~ -(I
- ~)~(a - am-)2 = 0.
(28)
The expression in the first member of Eq. (28) can be interpreted as a function F of a variable, F( x2 ), which has a real and positive zero for x2 = x2. On the other hand, it can be seen that the parabola F ( x2) has only a single real and positive zero, hence F( x2 ) > 0 for x2 > x2. In particular, if ( 1 + (T)~ > x2 (the
772
O.G. Calderh,
1. GonzalofOptics
Communicntkms
125
(I 996) 369-376
Axwc cmdi tion) , the function F takes a positive value for .$ = i I i CF)~. Hence, when x2 is replaced by i I - ~7)’ in Eq. (28) we have F(( I +a)‘) =4cr[~l$_cr)~+(~-~aar)2-r(t+lrr)21 > 0.
(29)
Thus the stability condition [ I -i- o-j2 > ,y2 is equivalent to the condition (29t, which can be rearranged to read
(35) is obtained. Then, for an electron-vibration interaction strength high enough, or for concentrations lower than a threshold one, Nthr, there is no laser emission for any value of the pump. (IS) The parabolic function in f 30) has two real zeros for a given value of the detuning - two pump boundaries - r+ and r”_, r*=(1/2&2){2Sacr+
fl>flV-
(1 +g)’
[2Satr+(l+(T)*]y$(1-t.a)*+S2 rt[(26ff~+((1+0+)~
> 0.
(30)
If the new parameter a, which depends on the vibrational coupling, is zero, condition (30) gives the wellknown first laser threshold f normalized pump ro ), A2 r <: I + (, +
c.f=O).
s ro.
a)2
(31)
In our case the condition (30) is a quadratic function of Y, which leads to the following CUSS: f A j The condition (30) is always satisfied for any value of the normalized pump Y. It means that the parabolic function in (30) has only complex zeros. For this to occur the condition for the coefficients of the parabola is > 0, which gives a condition L1:> cl~, S
(321
for the new parameter a
s+J&t(l+(T)* 2cT
a
(33)
The other zero of the function in the 1.h.s. of (32), a_, is negative and has no physical meaning. Then, under the condition ( 33) the trivial solution is always stable and laser emission cannot be obtained however much the pump be increased. Introducing a zz f/ (ecu), with cz in Nwl,u~,j*/(2~0cfiY_~), a condition for the vibrational coupI& strength f,
.f >
s+ J&iCN
(1 3
&cr
or for the concentration
+g_)* 1
N of the active centers
(341
-4(fl
+a)*es’)O”@Zj~~Z].
(36)
The condition (30) is then satisfied (the trivial solution is stable) only if r < r- or r > P+. If p- < r < r+ the trivial solution is unstable. In this case there is at least one oscillating solution for the field amplitude (although constant intensity), being similar to the solution of the CLE but with different period. For values r > T+, it was found numerically that other stable solutions in addition to the trivial one may occur; the trivial solution, the oscillating one. or both solutions coexisting, may take place depending on the initial condition. Concerning the stationary solution E,y= P,, lE,yI”= I”--I, D,y= 1, (which occurs only if ncT = 6) the stahility analysis leads to a linearized system of five equations (for E,, E,‘, P,,. P,* and D,) . The corresponding eigenvafue problem can he reduced by means of a straightforward algebraical manipulation to solve the cubic equation h”$(l+cr+y)A”t-y(l+cr+IE.~I’)/\ +2gyiE,.1* = a. The stability condition (I Ca+yfy(l
(37) is then
-W-i+/‘)
-2oy!Gj
>O, (38)
which is equivalent
to the two conditions
O.G.
Culderdn.
i.
Gonzalo/Optics
Communicuiions 125 (1996)
X9-376
24 ;23 z UI - 022
b
Fig.
I. ( a)-(d):
Decimal logarithm of the threshold molecular density versus A z (co12 - w)/y~, for different values of the vibrational f = 2 x IO I2, 6 x IO’*, 1013, 2 x IO”. (e)-(h): For each one of such values of f, the corresponding normalized
coupling parameter pumps. Y-
(thin solid line), Y+ {dashed line of long dash), the upper bound rUP (thick solid line), and rn (f
dash ). versus d. for molecular density N = IO2i m-‘.
= 0) (dashed line of short
The boundary between r.- and r+ sections is marked by a small horizontal segment.
where the solution lE,s12= r - 1 has been used. The first condition is a bad cavity condition; the second one mews that the pump must no exceed a given value.
4. Numerical
examples
Let us consider molecular data representative of some substituted aromatic molecules (for example), which are frequently used in lasers and nonhnear optics. Let we choose wt2 = lO’$ s-t (in the optical range) and the transition dipole moment of the order of the Debye ~~2 = 1 D = 3.3695 x lo-“O C m. We
,574
take y;’
O.C. Calderrin. I. C;onzulo/Optics C~lmrn~nicut~o~.~ 125 (1996) 369-376
= I W9 s, and a shorter value (as usual)
for the dephasing or decoherence time 7;’ = lo-” s. We also take N = l023 molecules rnw3, the length of the cavity L = 0.01 m, and the reflectivity of the mirrors R, = I, R2 = 0.3. The flight time of the photon into the cavity [lo], t,, then verifies C,/EO 2 tC:’ E -c In &f&/L = 1.806 x lOto s-l. Concerning the parameter f E us/u/*, the value of the vibrational frequency w, is between - 10” s-t (some bending modes in large plane molecules) and - lOI s-’ . The dimensionless parameter Iu/*, which invotves the nuclei displacement relative to their old equilibrium positions and the strength of the coupling, may take values from 1 to 20 approximately [ 111. We then take f of the order of lOI s-t or lOi s-l. The variables A z (~12 - u>/~J_ and N can be easily controlled experimentally. We then represent the threshold concentration N&, given by (35), versus A for different values of f (Fig. 1). For N = 102” me3 we can see in the graphics the detuning range where N < Nthr (case A), and N > Nthr (case B). The curve for N&r shifts towards positive values of A as f grows. The behaviour of the normalized pump ro, r+, and r_, as functions of A is also shown in Fig. 1 for each one of the values of f considered above. The wellknown laser threshold for f = 0, ro, is a parabola centered on A = 0. It is shown in the graphics for the sake of comparison when f #O. The fact that the population inversion per molecule induced by the pumping, da zz Da/N, cannot be physically greater than unity, leads to an upper bound for the pump given by rup E 2ctrt,, which is shown in the graphics. Values of I > ruP have no physical meaning. For low enough values of f, the curve r_ versus A approaches to that of ro, as expected, whereas r.+ > rup has no physical meaning. However, for greater values of f’, the values of P+ can be reached physically, and they are lower as f grows, at the time that the curves of T_+and Y_ shift towards positive values of A and slope to the right (Fig. 1e-h). That is, the eiectronicvibrational coupling shifts the region of laser oscillation to a lower frequency (higher value of A). The fact that the I+ section slops to the right allows that laser action be bounded by two pump values, r_ and r,_, at a fixed detuning (case B), so that for r > r+. laser action may disappears. Lasing occurs in the region bounded by r-, r+ and rup.
23.5
22.5
Fig. 2. Normalized pumps r_ (solid line), r+ (dashed line), bound TV,,(dotted line), versus decimal logarithm of
upper
molecular density N, for f = 2 x 10’” s-l
(Fig.
ih),
the same value of the rest of parameters as in Fig.
and the
3 = 30, and
I~
It is to be noted that for a given value of r, the population inversion per molecule induced by the pumping is do = r/rup. The dependence of the normalized pumps I_, J-+ and rupr on the concentration is shown in Fig. 2, for f = 2 x IO’” and A = 30. To calculate numerically the stationary solutions of the field in cases A and B, we choose f = 2 x IO’” (Fig. I h). For the data assumed we obtain c = 0.009 and y = 0.001. As an example of case A we take a detuning A = 5 (see Fig. Ih); then 6 = -15 and CI = 33.057. For any value of r < I^,~, the trivial solution (no laser emission) is stabIe. This has been the result using P = 10,15,20 < rg = 25.55 (no laser action for f = 0), as well as using r = 30,40,50 > r0 (laser action for f=O). As an example of case B we take A = 25 (see Fig. 1h); then 6 = 5 and a = 33.735, obtaining r- = 7.785, < r+ = 36.013 < rup = 65.654. Using r = 2,4,5,6 r_-, the solution found is also the trivial one for several initial conditions. For r_ < r < r+. (r = 10,20,30) the solution found is the oscillating one in the field amplitude. An estimation of the power, $~r~jE~j*lO-'~ MW cmv2, where EJ is related to E.y by ( 17), gives, for r = 30 ( /E,s/2 = 22.5), a value of 2.93 MW cmP2. Using r = 40,50,60 > r+, the trivial solution or (and) the oscillating one are found depending on the initial condition, contrary to the preceding cases. it is noticeable that for the same initial condition and changing r, we find the oscillating solution for Yclose
5. Conclusions
Fig. 3. The same as in Fig. 1for f = 2x lOJ3s-’ but now N = lo*’ m-‘, and cavity conditions leading to tc’ = 2.3 x lOI s-‘.
to r+, and the trivial one for greater values of r. This
would mean that the basin of the atractor for the trivial solution grows as r increases. Finally, we propose an example where the condition U(T = (0~12- w - f) /rl holds, which means that the total detuning including the dynamic one is zero, i.e., a new physical tuned situation is achieved. It corresponds to A = A, FZ c + f/y~_. In this case, the stationary solution E, = P,,, l&l2 = r- 1, D, = 1 is stable if the relations (39) are satisfied. For such conditions be satisfied with the same values of the molecular parameters, we change both the cavity and the molecular density. For the case f = 2 x IO’” s-’ , we take now L = 6x10-‘m,Rt =Rz=O.Ol (t,’ =2.303x101*s-I), and N = 10zs molecules me3 (see Fig. 3). It leads to P = I.151 > 1.001 = 1 + (q/y~), and rf = 31.808. The stationary solution occurs for A, = 20.387 (then 6 = 0.387, a = 0.336). As expected, this solution has been also obtained numerically using several values of Y < rf. The estimated power for r = 28 ( IE,y12= r - 1 = 27), e.g., is 3.514 MW cmF2.
Electronic-vibrationa coupling in the two-level active centers of a single-mode laser has been analyzed assuming a linear coupling between the excited electronic state and only one normal vibrational mode. This coupling leads to two new terms (one of them nonlinear) in the semiclassical Maxwell-Bloch equations. Such terms are proportional to the coupling strength and act as detuning terms. It can then be interpreted that the upper electronic level experiences a slight energetic shift proportional to the coupling strength, one of the contributions to this shift being also dependent on the population inversion. The result is that the electronic-vibrational coupling shifts the region of laser oscillation to a lower frequency at the time that the boundary of the region changes its slope. In addition to the trivial stationary solution (no laser emission) lZs = 0, P,7 = 0, D,s = r, the stationary solution of constant field amplitude, E, = Ps, /E,,1’= r- 1, D,y = 1, is found when the frequency of the field, w, is detuned with respect to the two initial electronic levels, 012, so that it compensates the energetic shift mentioned above and a tuned situation is achieved (ag = 8). Such a solution is just a modification of the well-known CW stationary solution for ,f = 0. In the stability analysis of the trivial solution, it is found that for values of the parameter n = f/( ecu) greater than the value given by (33) the trivial solution is always stable. Then laser emission cannot take place however much the pump be increased. This situation occurs, for example, in the case of high electronicvibrational coupling, or when the concentration of the active centers is lower than a critical vaiue Nthr given by (35). When the parameter a is lower than the value given by (33), i.e., the concentration is higher than N{hr, two boundary pumps, I_ and r,., given by Eq. (36). appear. It must be noted that r- tends to the well-known first laser threshold, r-0, as the electronic-vibrational coupling tends to zero. For the normalized pump values, r, lower than rl, the trivial solution is stable. For r_ < r < r+ the trivial solution is unstable and there is at least one stable oscillating field amplitude solution, similar to that of the CLE although with different period. For values of r higher than r+ the laser emission may disappear; the trivial solution is stable again but
i-6
O.G. Culderrin. I. Gonza6~/U~rics C~~muniru/if~ns 125 (1996) 369-376
other stable solutions may also take place. It is found that the trivial solution, the oscillating one, or both coexisting, can be obtained depending on the initial conditions. It could be understood that for high enough pump values (r > I-+), the global detuning, which depends on the population inversion, may be too high and the system be then too far from resonance. Further understanding on the behaviour of the laser system for values of r > r+ is the aim of a future work. Concerning the stationary solution of constant field anlplitud~, its stability requires a bad cavity condition and pump values lower than that given by (39). Finally we remark that similar results could also be found in other formally analogous situations where the excited state is coupled to other system as, for example, in the case of excimer formation.
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