Two-photon laser and optical-bistability sidemode instabilities

Volume 49, number 6

OPTICS COMMUNICATIONS

15 April 1984

TWO-PHOTON LASER AND OPTICAL-BISTABILITY SIDEMODE INSTABILITIES Shlomo OVADIA and Murray SARGENT Ill Optieal Seiences Center, University of Arizona, Tucson, AZ85721, USA

Received 5 December 1983

Two-photon laser oscillation and optical bistability have been observed and studied theoretically by various groups. The studies have not considered the buildup of sidemodes. We have extended the corresponding one-photon sidemode treatments to two-photon transitions for which no resonant intermediate levels exist. For the two-photon laser, we find a wide region of slngle-wavelength instability for a centrally tuned strong mode. For the corresponding two-photon absorptive optical bistability, we predict no single-wavelength instabilities. For strong-mode detuning, we find that appreciable regions of sidemode gain exist for both problems allowing multiwavelength instabilities to occur.

In recent years, two-photon laser oscillation has been observed in a Fabry-Perot cavity filled with rubidium vapor by Grynberg et al. [1] and in lithiumvapor cell by Nikolaus et al. [2]. In addition, twophoton optical bistability has been observed by Giacobino et al. [3]. Hence the stability of such nonlinear two-photon systems is of considerable interest. Theoretical investigations of the two-photon laser have been pursued by Walls and coworkers [4], and of the two-photon amplifier by Narducci et al. [5]. Two-photon absorptive optical bistability has been studied by Arecchi and Politi [6], Reid et al. [7], and others. However none of these analyses treats the buildup of sidemodes well known for single-photon lasers [ 8 - 1 0 ] and absorptive optical bistability [11]. In particular, in homogeneously-broadened lasers, if the cavity linewidth is larger than the sum of the dipole and population decay widths, the nonlinear polarization allows three frequencies with net gain to be simultaneously resonant for a single wavelength. This leads to sidemode buildup initiated by spontaneous emission [12]. Hendow and Sargent [13] showed that this property of the polarization is due to population pulsations, i.e., due to the response of the medium to the adjacent mode beat frequency. Casperson showed [14] that this property could also occur in purely inhomogeneously broadened media due to spectral-hole burning, This sidemode instabili0 030-4018/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

ty has been observed [15] notably in He-Xe lasers, in which typically a combination o f the populationpulsation and hole-burning mechanisms play a role. These instabilities have even led to the period-doubling route to chaos [16]. Sidemode buildup with each mode having a different passive cavity resonance (multiwavelength instability) was also predicted and has been recently observed [17]. In absorptive optical bistability, single-wavelength sidemode buildup cannot occur, but multiwavelength instabilities have been predicted [11]. In another paper [18], we calculate the complex two-photon polarization for one, two, and three wave interactions in steady-state. We use these polarizations in the present paper to find a sidemode dispersion relation for the two-photon laser and optical bistability. Applying the H'urwitz criterion [19] for central tuning of the strong mode, we derive conditions for the onset of single-wavelength laser instabilities. We illustrate numerically the sidemode instabilities that occur for strong mode detuning. Finally, we show that multiwavelength instabilities occur in both the laser and optical bistability. This paper is organized as follows: First, we briefly summarize the onephoton instabilities for the laser and optical bistability in homogeneously broadened media. Second, we consider the two-photon sidemode instabilities for these systems. 447

Volume 49, number 6

OPTICS COMMUNICATIONS

E~ EI D/ /

15 April 1984

I 0

]

i

Er

E2

,<

Et

E

Fig. 1. Unidirectional ring cavity geometry. EI, ET, and E R are the incident, transmitted, and reflected fields, respectively, for optical bistability. E I and E R are not present in our laser analysis.

We consider the unidirectional ring cavity geometry of fig. 1, in which mirrors 3 and 4 are assumed to have 100% reflectivity, and we ignore all the time delays. For the laser problem, the steady-state oscillation condition requires that the saturated gain equals the loss. In the one-photon homogeneously broadened medium, the gain saturated by the strong-mode dimensionless intensity 12 is given by a = aoL2/ (I + I2L2). Here 12 = [E212/E 2, E 2 is the strong mode comj~lex amplitude, E 2 = (~t/~)2'yTii, the lorentzJan L 2 = -rz/[7 2 + (co - u2)2 ], and the linear absorption/gain parameter is s 0 = K~2N/h3,eo , where ATis the average unsaturated population inversion. This yields the steady-state intensity 12 = N - 1, where N is the relative excitation. For the absorptive optical bistability, 12 is determined by the state equation

EI/'T 1/2 = ET/T1/2(1 + od/2T),

(1)

where E I is the input field, E T is the transmitted field related to 12 by (ET/T1/2Es) 2 = I2, and c~ is the saturated absorption. Given the steady-state intensities 12 for these problems, we ask, can sidemodes build up? To answer this question we assume a three-mode, slowly varying electric field in the form

E(z, t) = ½ [El(Z)exp(iAt) + E2(z )

I

60/2

Fig. 2. The sidemode diagram with the central strong-mode tuning.

lation matrix in the adjacent mode beat frequency, thereby converting the Schr6dinger equation of motion to a set of difference equations. Furthermore, we derive analytic expressions for the Fourier coefficients to first-order in the sidemode amplitudes. Coupling these polarizations with Maxwell's equations, we find the steady-state dispersion relation

~'n -- f2n + iK = -(Vn/2eo)Pn/En,

where K = u/2Q is the cavity damping constant, un and E n are the mode frequencies and complex amplitudes, respectively, and f2n are the passive cavity frequencies. The un are assumed to be equally spaced. This dispersion relation applies to both the laser and optical bistability problems. The corresponding sidemode polarizations differ in their overall signs (gain versus absorption) and in the equations that determine the steady-state strong mode intensity 12 . Consider now the stability of the two weak sidemodes. Specifically, we insert the identity u 1 - ~21 = u l - co - (g21 - co) = --A -- 6, the steady-state laser intensity 12 = N - 1, and the sidemode polarization P1

a0

1 -12 +iv

- 2 e 0 E-]-=i I----~ 1+ (1 + i x ) ( l + i y ) + I 2

(2)

where E l and E 3 are the two weak sidemode amplitudes shown in fig. 2. Note that iA is a self-consistent complex eigenvalue that includes all exponential time variations of the sidemode amplitudes. These assumptions allow us to Fourier analyze the medium's popu448

(4)

into the dispersion equation (3). Here y = A/711, and x = A/% This gives the cubic eigenvalue equation X3 +C2 x2 + c l ~ , + c 0 = 0 ,

+ E3(z ) exp(-iAt)] exp(-iv2t ) + c.c.,

(3)

where c O = 2K"/"/III2 + ir53,3',(1 +/2), C1 = (1 + I2)'}'3', + KT, + i~5(3'11 + 30, c 2 =K +3,tl + 7 +i6.

(5)

Volume 49, number 6

OPTICS COMMUNICATIONS

This is the same result as ref. [8], eq. (3.7) provided one substitutes k ~ -~3, 6 -+ a, and 12 -+ X in eq. (5). Nonzero 6's lead to the multiwavelength laser instability [10,11]. For the one-photon absorptive optical bistability one can take s 0 = - 2 ~ C in eq. (4), where C = o.L/2T. In a derivation similar to the laser case, we combine the intensity 12 determined by the optical bistability state equation (1) with the polarization (4) to get the eigenvalue equation (5) with the following coefficients:

I_

1 - 12- ]

(6)

This result is the same as the one derived by Bonifacio and Lugiato [17]. Note that for an absorbing medium, sidemode gain occurs for detunings smaller than the Rabi flopping frequency, while for gain media, sidemode gain occurs for a detuning about equal to the Rabi flopping frequency. We now consider two-photon transitions for which no resonant intermediate levels exist. This case allows one to use the two-photon two-level mode [20,5] as shown in fig. 3. Two basic differences occur between the one and two-photon models. First, the two-photon coherence Pab induced between the levels yields a polarization only with the help of an additional atomfield interaction; second, dynamic Stark shifts of the

L

o~= %I2L2/(1 +I~L2) ,

(7)

(K/2eo)N[ Kab 1(711/7)1/2, the lorentzian L 2 = 1/ [1 + (co - 2v2)2/72], and the steady-state intensity

711')'(1 + 12) +t~ IT + 7tj + 1~/~]g. + i6(7 It + "/),

c2 = • + ")'It + 7 + i6.

level frequencies that are neglected in the one-photon case can play an important role in the two-photon case. These differences complicate the analysis substantially. For the simplicity of the present paper, we assume that the dynamic Stark shifts vanish (co s = 0), and that the level decay-rate constants are equal (7a = 7b)- For a two-photon homogeneously broadened medium, the sidemode saturated absorption is given by

where the two-photon absorption parameter is a 0 =

Co = •T)'II 1 + I 2 + 2 C 1--~2J+i87117(1 +12),

¢1 =

15 April 1984

j I j)

Fig. 3. Two-photon two-level scheme. All intermediate levels j are sufficiently nonresonant that they acquke no appreciable population.

is 12 = IkabE2 [(T 1T2)l/2/2h. From ref. [ 11 ], the sidemode polarization is v v Xa + ~'b i°~0"YI2 2e 0 P1 = - ~ e kaa - E1 + - -

7tt

1 + I2L 2

Elf 1 +E~]'~

X [2D1E1 + D2E~ -D112 1 + I ~ ) - ~ l l + D ~ ) / 2 z

1

(8) where (with m = 0, -+1)

D2_m =

[7 + i(co - 2u 2 + m A ) ] - l ,

f l = 7(D1

+D~)F(A),

f~ = 7(D 2

+D~)F(A),

F(A) = (1 + iA/Tii) -1. The first term of the right-hand side of eq. (8) is just an overall frequency shift and therefore can be ignored. The D2E~ term results from the wave E 3 scattering off of the Pab induced by interactions with the strong mode alone. This term does not appear in the one-photon case, and is comparable in size with the other terms. The steady-state unidirectional ring laser intensity is 12 = N + (N 2 - 1) 1/2, which implies a firstorder phase transition in contrast to the second-order one-photon case. Using this intensity and taking co = 2v2, we find the dispersion relation for the sidemode E1

1 +iy-I 2 -&-

(9)

6 +iK = iK + 2iK (1 + ix)(1 + iv) + 12

449

Volume 49, number 6

OPTICS COMMUNICATIONS

Substituting X = iA into eq. (9), we obtain a cubic eigenvatue equation (10)

~.3 +c2~2 + c i r . + c 0 = 0 , with the complex coefficients CO = TTII[2K(I2 - 1) + i8(1 + i2)], c 1 = 3'TIL(1 + I 2 ) -- 27K + iS(T + 711), c2 = 3' + Tll + i8.

However, for central strong-mode tuning and singlewavelength instabilities, 6 = 0, which gives real coefficients. This allows us to use the Hurwitz criterion to predict the occurrence of positive real parts of the eigenvalues, i.e., the onset of sidemode instability. We find from the instability condition ClC 2 - c O ~ 0 that single wavelength instability occurs when I2(2K - 7 - V,) > 7 + %1 - 2KT/%I.

(11)

This gives a number of instability regions. In particular if T = VII, the eq. (11) reduces to I22(g - 3") > 3' - ~¢.

(12)

If 3' > K, this inequality is never satisfied, and therefore all values of 12 are stable. Conversely, if 3' < K, all I 2 are unstable. For two-photon absorptive optical bistability, 12 is determined by a nonlinear algebraic steady-state equation, and the sidemode dispersion relation is given by

1 +12 (1 +ixX1 + iY) + I 2

(13)

Eq. (13) can be written as a cubic eigenvalue equation ~k3 +c2~, 2 +Cl~ + c 0 =0,

(14)

where

I

2KCI2

CO =')'Tit (I +I2){K + i8) +1--~/2 (3 - I 2

)1

,

2gCI2(Ti I + 37) c I = 7711(1 + 1 2 ) + ( 7 + %I)(K + i8) + 1

c2 = T + TII + i6 + g[1 + 2CI2/(1 + i2)]. 450

Setting 6 = 0 for central strong tuning and singlewavelength instabilities, and applying the Hurwitz criterion reveals that the two-photon bistable system is stable except for the field values in the negative slope region, which decay to the lower branch. So far in our analysis, we have assumed central tuning of the main mode, and that the pair of sidemodes act in phase with one another. If one allows strong-mode tuning, the sidemode amplitudes are unequal and the simple dispersion relation (eq. (3)) is inadequate. Instead, following ref. [21 ], we have a coupled-mode equation diElt=

( al--~'/2Q1 +i(" I - ~ 2 1 )

dt\E~]

\-iK 3

+I 2

iKT )(/Z'l)

oe;-u/2Q3-i(u3-n3)/\E;/'

(is) where Qn is the cavity Q for the nth mode, the complex absorption coefficient

c~ 1

-

a°I2 [2 _ TD 1 1 + t2L2

_

_ 7DlI2F(A)

T(D1 + D~)

..~

(16)

1 +I2F(a)T(D1 + D~)/2J' and complex coupling coefficient

_i~ T -

"o'r~ [TD2

1 + I2L2 L

7(D_.~2 - TDII2F(

- 2 C I 2 (1 + iyX 3 + ix) - 12 K +i(A+5)=g

15 April 1984

)

+A~2

q

:1.

1 + I2F(A)7(D1 + D~)/2 A

(17)

Notice that Re(a 1) > 0 corresponds to gain in an inverted medium, while Re(a 1) < 0 corresponds to gain in an absorber. A single-mode laser is unstable if a resonant cavity sidemode sees a net gain in the presence of the oscillating mode. To inspect this condition, we diagonalize the coefficient matrix for the sidemode coupled-mode equations. For the single-wavelength case, we set £21 = g]2 = g23, Q1 Q3 = Q, and K = u~ 2Q, which gives the following eigenvalues TM

7Vl,2 = -• - iA + (t 0, Im(~.l, 2

+

iA)/K

=

(19)

--(u 1 --

(20)

u2)/t