A new single-mode laser equation

Volume 145, number 4

PHYSICS LETTERSA

9 April 1990

A NEW SINGLE-MODE LASER EQUATION Li CAO and Da-jin WU Department of Physics, Huazhong University of Science and Technology, Wuhan, PR China Received 6 December 1989; accepted for publication 5 February 1990 Communicated by A.R. Bishop

A new complex field amplitude equation for the single-mode laser with several colored multiplicative noises is derived. The relation between our equation and the single-mode dye laser equations with gain noise is discussed. To study the statistical properties of the single-mode laser using the equation we obtained, an effective Fokker—Planck equation (EFPE) in which the laser intensity is decoupled from the phase variable is derived.

1. Introduction The role of multiplicative noise in nonequiibrium systems has been the subject of great interest in recent years [I]. The dye laser seems to have become the prototype ofnonequilibrium systems in which multiplicative noise plays an important role [2]. A stochastic theory with multiplicative pump noise seems to give an adequate account of the statistical properties of the dye laser [31.In these theories, however, the strength and the correlation time of the noise are treated as adjustable parameters, and the source and the nature of multiplicative noise in the dye laser are not well characterized [4]. The complex field amplitude equation for the single-mode dye laser with colored noise is still phenomenological. In this paper, we derive a stochastic differential equation of the complex field amplitude of the single-mode laser from the set of semiclassical single-mode laser equations in terms ofa new adiabatic approximation which was developed recently by Schoner and Haken [5]. The equation we obtained contains naturally several colored multiplicative noises, and includes the complex field amplitude equations of a single-mode dye laser with colored gain noise [6] or white gain noise [7] as the special case. In the new equation, the source and the nature of multiplicative noises are clear, and the strength and correlation time of the noises are determined by the system itself. The relation between our result and the previous stochastic theories of the single-mode laser is discussed. Finally, the corresponding EFPE is obtained by means of the projective operator technology and a Hanggi-like ansatz which the authors have used to obtain an EFPE from the multidimensional non-Markovian processes [8,9].

2. Derivation of the complex field amplitude equation for the single-mode laser Our starting point is the set of well-known semiclassical single-mode laser equations b=—K6—iw5—i~g,~&,,+P(t), =—yã0—ii+ig&~+P0(t)

(1) ,

(2) (3)

0375-9601/90/$ 03.50 © Elsevier Science Publishers BY. (North-Holland)

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9 April 1990

where /‘ and F* are time dependent dimensionless complex field amplitudes. We have distinguished the atoms by the index u, and confine the treatment to the two laser active atomic energy levels. ~,, is the complex dipole moment in dimensionless units of atom ji, and a,, the inversion. With the help of the “quantum—classical correspondence”, the amplitude B, the dipole moment a,, and the inversion a,, have been treated as classical quantities. w is the mode frequency, ~ is the central frequency of the atom (assumed to be homogeneously broadened), K is the decay constant ofthe field mode if left alone in the cavity without laser action, y is the linewidth of the atom caused by the decay of the atomic dipole moment, ~ is the relaxation time after which the inversion comes to an equilibrium, d0 is the equilibrium inversion which is caused by the pumping process and incoherent decay processes if no laser action takes place, and g,, is a coupling constant describing the interaction between the field mode and the atom p. g,, is proportional to the atomic dipole matrix element. For running waves in a single direction the coupling constant g,, has the form g,,=gexp( —ikx,,). g is assumed real. F. P,, and Ta,, are the fluctuating forces. We assume exact resonance. w= p. and eliminate the main time dependence by the substitutions 1 P(t) =F(t)e r,,(l) =T,,(t)e b=he a,, =a,,ev Eqs. (l)—(3) then read .

~‘

,

~

h=—Kh—i

~g,,a,,+F(t),

(4)

=—+ig~,,ba,~+T,~(t)

(5)

.

&,,=y(d 0—a,,) +2i(g,,ct,,b*_c.c.)+T’,,,(t)

(6)

,

where F(t), F,,(t) and E~,,(t)are assumed to be Gaussian white noise. To eliminate the atomic variable a,,, we assume that the conventional adiabatic approximation holds at this step. So we get from (5) a,,=i~ba,,+ ‘F,2(t).

(7)

Substituting (7) into (4) and (6), we have h=—Kb+~b~a,,+P(t), a,,=~’~ (d,~—a,,)

—

bb~a,,+

(8) [b2F,,1(t)—b1T,,2(t)] +f~,,(t),

where t(t)=—i~g,,F,,(t)+F(t).

T,,=F,,1+iF,,2.

2/y. B=4g2/y, C=(4g2/y)d

Let a~=a,,—d0,A=zg Ag= be the gain, and a= ~DoKrAgK y the net gain, then (8), (9) become 160

0, and

(9)

Volume 145, number 4

PHYSICS LETTERS A

9 April 1990

b=ab+Ab ~ a~+P(t), p

(10)

&~=_yiia~_Bbb*a~_C’bb*+ [b

2F,,1~1(t)—b1F,,F~2(t) ] +F0,,,F~,,,(t).

(11)

In eq. (11), we have written the noises E,,~(t), I’,,~(t), and Fa,,,(t) as F,,F’,~I(t), F,,V~2(t),and Fa,pI~y,,,(t) respectively. They are defined by <[‘,,,(t)F,,1(t’ [‘~,p=T’a,p1

)> =2F~ö.~,ô(t—t’) =2F~,,,ö~(t_t’) ,

(i,j= 1, 2),

+iT’,,,,2

In this paper, we used completely dimensionless equations by scaling both the time and the complex field amplitude. That is, we have chosen the scaling factor of time as l0~ s as in ref. [6]. The definition of the dimensionless b is chosen as in ref. [10]. Now, we must use the new adiabatic approximation, the stochastic generalization of the conventional adiabatic approximation [5], to eliminate the variable a’,, from (10). To this end, we assume 2, y 2. b~ö, a—~ 11~A-~B~C’~ô°, Fp~Fap~ö Here ~ is a small parameter. We expand a~as follows, ~ C~.

(12)

m~2

According to ref. [5], C~ are given by

C~=

~

di,

(13)

where, in our situation, P

2)

0= _Bbb*a~,,_C~bb*=~

p~m)

=_Cbb*,

p~3)=0,

P~m)(m~4)= _Bbb*C~2),

p~

F,, 1=~b2F,,= ~ ~ g

,n=2

F,,2=—~b1F,,= ~ F~, g m=2 Fa0

~ ~

~

Fft=~b2F,,, g

F~(m>2)=0,

F~~=—~b1F,,,F~(m>2)=0, g F~(m>2)=0.

,n=2

m. The subscript For the superscript (m) denotes that quantities are ~ô “ad”the in aforementioned C~ denotes thequantities, new adiabatic approximation described in those ref. [5]. Thus,

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9 April 1990

C~= ~

~bb~+

~b2Z,,1(t)— ~b1Z,,2(t)+Z~,,(t).

(14)

According to the theory of Schoner and Haken, the integral operator JL~ exp[ erate on the variables b, b*, b1 and b2. In eq. (14), we have denoted

J J

Z,,1(t)=

Z,,2(t)=

Z~,,(t)=

—

~

(I—i)

I

...

di does not op-

exp[—y11(l—z)]F,,F’,,1(i)di,

(15)

exp[—y11(t—t)]F,,E~2(z)di,

(16)

exp[—yj1(t—i)]F~,,T~,,(i)di.

(17)

We find C~=

J

4~di= exp[—y11(t—i)]P~

J

exp[_yij(t_i)](_Bbb*)C~di

—

=

(_ ~~~~bb*)(_ ~bb*)+(_Bbb*)(~b

2~-~z,,1 g

—

y

~b1

~-~z,,2+~

g

y

?ii

and C~)=

(—

~bb*)(_

~bb*)+

where the operator G~)=y~~J~L exp[ In addition, C~—0 ad —

(_Bbb*) (I—i)]

—~

[~b2(~)Z,,1 ...

—

di.

(2k+I)_ ad —

‘

So, we find from eq. (12),

~ c

)=(_f~_bb*)

k=l

g

~ k=1

(_~bb*)~+~b2

g

/ii

(_-~bb*G~j~’z,,i ?ii

~ (_~bb*G~)~’z,,2+ ~ (~bb*G~o)~Zap. k

I

lu

k

I

Substituting (18) into (10), we obtain 162

~ k=l

hi

(18)

Volume 145, number 4

PHYSICS LETTERSA

B

A

9 April 1990

Ab

b=—Kb+ l+(B/yui)bb*gb2l+(B/yuu)bb*G~1 bi I + (B/yii)bb*G~Z2 + 1 + (B/yui)bb*G~Za+t,

—

(19a)

where

~

Z2= ~ Z,,2,

Z,,1,

/L

Z~=~ Z~,,,.

(19b)

/L

Let E=\/~7~7b, we obtain the complex field amplitude equation for the single-mode laser from (19a)

(1= EE*): E=-KE+

l~’

~1E+2~yE2

For large ~

Z1—2~yE1 l+IG~Z2+ l±IG~ Z~+~7~7F.

(20)

_2~J~7~E1 ~

(21)

we have

G~Z1Z1, G~Z2Z2, G0ZaZr,. So, eq. (20) becomes

E= -KE+ ~-1E+2\/~77yE2~ZI

+ ~-jZa+~/~7~Tt.

The statistical properties of the Ornstein—Uhlenbeck (OU) noises are as follows, =0,

=ö~exp(— It—t’ It)

=O, ~

(i,j= 1,2),

(i,j=l,2),

(22)

where

1~L, ~ Yuu

~ Yuu

p

ZaZ,ii+~Z,y2. Yuu

p

3. The complex field amplitude equation for the single-mode dye laser with multiplicative colored noises Eq. (21) is essentially a complex field amplitude equation for the single-mode dye laser with several multiplicative OU noises. The reasons are as follows: (1) Though our eq. (21) was based on the following assumption: the dye active medium can be described by a homogeneously broadened two-energy-level system, this assumption is consistent with the characteristic of a dye active medium. We know that the dye laser is a typical homogeneously broadened laser system, and the conditions of operation of the dye laser indicate that the triplet state effect is not dominant since a triplet quenching agent was used in the dye solution. So the dye laser can be modeled by a two-active-energy-level system, except for the phenomena which Hong Fu and Haken have discussed recently [11]. (2) The organic dyes are large and complicated molecular systems containing conjugated double bonds. An important characteristic of these dyes is that they have a very large dipole matrix element, therefore the coupling constant g is very large too [121. This leads to an important result: for the dye laser, the constant A in eq. (21) is much larger than the non-dye one. Thus, for the dye laser, we expect that those multiplicative noise 163

Volume 145, number 4

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terms which associated the OU noises Z1, Z2 and Z~may have some observable effects. However, for the nondye laser, these terms may be neglected because the constant A is very small. (3) The colored gain-noise model for the dye laser fluctuations [6] can be seen as a special case of eq. (2 1 when we neglect the noise terms associated with Z1 and Z2. In addition, when we take furthermore the white noise limit of the OU noise Z,,, the white gain-noise model is obtained [7].

4. The effective Fokker—Planck equation We derive an EFPE from the stochastic differential equation (21). To this end, we write the complex field amplitude E in the form E= E~e~=~/Ie~’. so that eq. (21) can be written in terms of a pair of stochastic differential equations for the laser intensity I and the phase up: 7J7y sin up Z 811= 21( K+ + 4~J~ 1 4\/~T~1/y cos ~ z2 + 41 77~ItIcosup+2\/7yIlF +2~’~ 2sinup, (23) ~

—

ö~up=~Za2

—

+~7~1IF2cos up—~B/y11IF1sin up,

where Z,,=Z~1+iZ~2,f’=P1

(24)

+iF2.

To obtain an EFPE for (23) and (24). we consider the general form of the n-dimensional Langevin equations, q,,=.f,,(q)+g0~(q)~~(l)(p= 1, 2 where

~

n

1,2

M)

I—ti

/i~).

p=

(25)

.

are OU noises given by 2~exp(

—

<~,>=0.

<~(t)~(t’ )> =ô~

(26)

We have developed a projection operator technique to obtain an EFPE from eqs. (25) and (26) under the condition that rD is small [8]. The EFPE corresponding to eqs. (25) and (26) is [8] 0,P(q, I) = _8q~fp(q)P(q,1) +8q~g,,p(q)ö~D~[ 1 where

iv0q,~f,i

(q)

(q),

~

(27)

the factor D~[l—i~Bq~,f,

1(q)]qq, means that q takes the value of the steady state q~of the deterministic of eq. (25), i.e., q~satisfies f,,(q~)=0 (p=l, 2 n), which is the result of the Hanggi-like ansatz [9]. The EFPE corresponding to eqs. (23), (24) and (22) for the probability density function P(I, up, I) is

parts

0,P(J, up, t)= _01[21(_K+

~I)_2\/~~7D~1P(I,

up, I)

+ ~

312 Al312 +l6~-~X4D[l_2Ki(K/Ag_l)]8ij~j0i Al 1~j (28)

~

This equation is obtained as follows: let q1 and q2 in eq. (25) be land up respectively, let f~(q)and J~,(q)in eq. (25) be 21[ —K+Ag/(l +1)] and 0 respectively; and let ç~, ~ and ~6 in eq. (25) be 4~.Z,,2, F1, ~.

164

~,

~

Volume 145, number 4

PHYSICS LETTERS A

9 April 1990

F

2, Z1 and Z2 respectively, then eq. (25) for two dimensions (n = 2) and for M= 6 becomes eqs. (23) and (24). Using these new representations, we obtain directly eq. (28) from eqs. (22)—(24) and (27), where DL. is the intensity of P1 and F2. From eq. (28), it is possible to obtain a stochastically equivalent model to (21) in which the intensity is decoupled from the phase variable:

011=21(_K+ ~ +2{D, 1[l

_2~D~)+2(B/Y1L)u/4DU2fIP~(t)

312 1”2j~j[”~i(t) Al Al 2K1(K/Agl)]}h/21 7F~,,,l(t)+8\/’~7Ty{D[l_‘2Ki(KIAg_l )]} 1/2

8,up=2{D~[ I _2Ki(K/Ag

1) ]} u/2(~~)

2P~(t), (D~/l)1/ where the aforementioned equations were obtained according to the Stratonovich interpretation and E~,,,

(30)

2(t)+~

=ô

1ô(t—t’ )

,

(f~,(t)F~1(t’ ) > =ö0ô(t—t’

=~,,ö(t—t’)

(i,j= 1,2)

.

) (31)

References [1] W. Horsthemke and R. Lefever, Noise-induced transitions (Springer, Berlin, 1984); A. Schenzle and H. Brand, Phys. Rev. A 20 (1979) 1628. [2] K. Kaminishi, R. Roy, R. Short and L. Mandel, Phys. Rev. A 24 (1981) 370; R. Short, L. Mandel and R. Roy, Phys. Rev. Lett. 49 (1982) 647; P. Lett, R. Short and L. Mandel, Phys. Rev. Lett. 52 (1984) 341; R.F. Fox and R. Roy, Phys. Rev. A 35 (1987) 1838; P. Jung, T. Lieber and H. Risken, Z. Phys. B 66 (1987) 397. [3] R. Roy, A.Z. Yu and S. Zhu, Phys. Rev. Lett. 55 (1985) 2794; S. Zhu, A.W. Yuand R. Roy, Phys. Rev. A 34 (1986) 4333; R. Graham, M. Hohnerbach and A. Schenzle, Phys. Rev. Lett. 48 (1982) 1396; F. de Pasquale, J.M. Sancho, M. San Miguel and P. Tartaglia, Phys. Rev. Lett. 56 (1986) 2473, and references therein. [4] R.F. Fox, G.E. James and R. Roy, Phys. Rev, A 30 (1984) 2482; Hong Fu and H. Haken, Phys. Rev. A 36 (1987) 4802; M.R. Young and S. Singh, Opt. Lett. 13 (1988) 21. [5]G. Schoner and H. Haken, Z. Phys. B 63 (1986) 493; 68 (1987)89. [6] E. Peacock-Lopez, F. Javier de Ia Rubia, B.J. West and K. Lindenberg, Phys. Rev. A 39 (1989) 4026. [7] M. Aguado, E. Hernandez-Garcia and M. San Miguel, Phys. Rev, A 38 (1988) 5670. [8] Li Cao, Da-jin Wu and Hui-xian Wan, Phys. Lett. A 133 (1988) 476. [9] P. Hanggi, T.J. Mroczkowski, F. Moss and P.V.E. McClintock, Phys. Rev. A 32 (1985) 695. [101 H. Haken, Lasertheory (Springer, Berlin, 1984). [11] Fu Hong and H. Haken, Opt. Commun. 64 (1987) 454; Phys. Rev. A 36 (1987) 4802; Opt. Soc. Am. B 5 (1988) 899; Phys. Rev. Lett. 60 (1988) 2614. 1121 0. Svelto, Principles of lasers (Plenum, New York, 1982); F.P. Schaefer, ed., Dye lasers (Springer, Berlin, 1973).

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