EFFECT OF CAVITY AND ATOMIC "DETUNINGS" ON STABILITY IN A NEW FORMULATION OF THE STEADY STATES OF THE RING LASER WITH A SATURABLE ABSORBER

(1)

HAL Id: jpa-00227703

https://hal.archives-ouvertes.fr/jpa-00227703

Submitted on 1 Jan 1988

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

EFFECT OF CAVITY AND ATOMIC ”DETUNINGS”

ON STABILITY IN A NEW FORMULATION OF THE STEADY STATES OF THE RING LASER WITH A

SATURABLE ABSORBER

D. Chyba

To cite this version:

D. Chyba. EFFECT OF CAVITY AND ATOMIC ”DETUNINGS” ON STABILITY IN A NEW FORMULATION OF THE STEADY STATES OF THE RING LASER WITH A SATURABLE ABSORBER. Journal de Physique Colloques, 1988, 49 (C2), pp.C2-367-C2-370.

�10.1051/jphyscol:1988286�. �jpa-00227703�

(2)

JOURNAL DE PHYSIQUE

Colloque C2, Suppl6ment au n06, Tome 49, juin 1988

EFFECT OF CAVITY AND ATOMIC "DETUNINGS" ON STABILITY IN A NEW

FORMULATION OF THE STEADY STATES OF THE RING LASER WITH A SATURABLE ABSORBER

D.E. CHYBA

Physik-Institut der Universitat Zurich, CH-8001 Zurich, Switzerland

Al>straet

T h e method is orttlined, grapl~ical resolts are presented, ancl laewly- discovered branches of zero-intensity sol~ttiolts are also discussed.

1 Introduction

This report sinlmlarizes several new results for a se~niclassical model of t,he detunecl ring laser with an intracavity saturable absorber (DRLSA) in the uniform field approxinlation. These are: (i) forn~ulat~ion of a set of ordinary differential equations for the cletuned case, by illearis of the uniform field approximation; (ii) an explicit solut.iou for t.he steady states of these in the case of non-zero int,ensity via the roots of polynonuals up to quartics; (iii) a separate forlnulation of the steady states for the case of zero intensity, leading to tho cliscovery of new branches of zero-intensity solutions even in the tuned case; and ( i v ) selected numerical solutions for the stability of the steady state solutions, showing the effects of cavity detuning. A cot~~plete and detailed presentation of this work is being prepared for pul~lication elsewhere.

2 Discussion

The niodel is based on the Maxwell-Bloc11 equations for a hotnogencously

,I

broadened DRLSA in whidi the media consist nf two-level atolns. 1 . 1 1 ~ results for non-zero intensity illustrated here were obtained by t,ransfornung the Maxwell-Bloch ecpations into a purely real set of partial ~li1rttrentiaI equations and applyillg the tmifonl~-fi~ltl a p l ~ r o x i ~ ~ ~ a t i r ~ n to I-~btain a set ol' ordinary differential equations which hold for the detuned chse. It has been possible to solve for the steady states of these; in the li~!ly tuned case, one has quadratic equations, and in the detuned case, a quartic in one of the state variables. Such a solution is, in principle, explicit, and allows very rapid and precise computation of the steady st,ates. The differential equations also provide a convenient foundation for a linear stability analysis of the steady states, sonie of the results of which are illustrated here. The effect of even a small det~uling on the solutions is evident.

A previous approach to this problem by the author and coworkers [I]

utilized a dispersion curve-mode-line approach due to Casperson and Yariv to obtain numerical solutions for the steady states. That 111ethod can also be applied in the case of t.he zero-intensity steady state solutions, since the dispersion curve still exists in this case, and depends upon the excitation of the medium. Such a method yields the previously known solution a t zero frequency and zero intensity for the tuned case, but also yields two adclitional branches, which bifurcate to form the known solutions at non-zero intensity.

Previously, these had been thought to appear with increasing excitation disconnected from other branches of solutions [2]. The niethod also allows one to find the corresponding branches in the detuned case. Finally, a stability analysis based on the corresponding differential equations can he made, showing the changes of stability at the various bifurcation points lying along the branches.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988286

(3)

Exclt'n A in units of Isa Fig,! Intensity vs. amplifier excitqtion in the fully tuned case. Solid lines: stable solutions; dashed, unstable. Both axes have units of the amplifier satura- tion intensity. Cavity linewidth: 0.40

YL4,

the polari- zation decay rate of the amplifying medium. These results agree with those previously described in Refs. 2. The straight stable segment at low intensity represents two branches degenerate in intensity but having different operating frequencies; compare Fig. 2. The degeneracy is broken by detuning, and the stability behavior is also changed; compare Fig. 3.

Q

LL

'70.0 I. 2. 3. *. 5. 0 E~cit'n Q in units of Isa Fig. 2. Operating frequency vs. amplifying medium excitation for the case of Fig. 1. The frequency is shown as displacement from the amplifier line center and is given in units of

&,.

Solid lines are stable; dashed, unstable. Zero-intensity solutions are the broad parabolic arc at left and the line at zero frequency at left. The latter continues to all higher A values but is unstable. Notc the isolated stable arcs surrounded by unstable arcs. Finite-intensity solutions are the narrower parabola and the right-hand line at zero intensity; compare Fig. 1. The finite-intensity parabola bifurcates from the zero- intensity parabola where the latter chanzes stability.

(4)

OE'O SZ'O OZ'O SL'O OL'O S 0 ' 0 0 '0

R S ] JO sJ!un u ! paJPnbS-apnl!lw-3

c ' w 5 fiA.2'" 8 .

- d w r c L I m .4 .4

85: a,.:*;

G a,

5 2Euh;;l

= g "'Z

>

h u m e c w o U

.

^C^{M U . 4}

.4

4.: .z .a :

530

$2, 5 .t:

In M O U

2 8 :

."$I

" ?

2 " "

4.c ³

4 - 3 0 1 P U 0

L1 o m d

P M u n

c h m m d h m .4 rl

c r l m - u

&,z ^; ^8-25

- 4 w u O k . 4 F q a e u U

a, a, c.4

0 h > . O . d W u u a , w r$.s.:<:

m m u

r l u o u u .4

f i a 2 . 5 A m

. d G P O U a , m m

d h d P

.

^,ⁱ^.^:

."d

';:

2 *

m . 4 m m V l C C U G . \ o a, 3.: s m o u

.d . ~ u u m

w 0 . d m u.4

I'

5

M S

c u 8 A

m h G C W S P 0

s o u .d

U A U

0

h5.z

^J

a u a

m m m . 4 2 r r G u

.z $ 2 2

$I

? a G m M ..A 3

U ~ M Io a.4 N 4 m w m

.4

u . 4 u u 3 M m .4

P . 5

G W W w o w

^ U U -_{m d}

.

u^.4G^.4_.4h^U2_u - 4 a 3 4 m

& m 0.4

dn a,

o m m d

4 J S U U P u m m m U m a m c m

74 u m . 4 G .4 0 .c

EiZ o m

'

- 4 G e m m _tinu

-

m o . 4 ^r^.4_u

. m p m m

M q ^m^U^h

.4 u r l m

:I--i 0 m RI ,G

(5)

C2-370 JOURNAL

DE

PHYSIQUE

References

1. D. E. Chyba, N. B. Abraham, and A. M. Albano, Phys. Rev. A 35, 2936 (1987).

2. (a) T. Erneux and P. Mandel, Z. Phys. B 44, 353 (1981); (b) 44, 365 (1981); (c) P. Mandel and T. Erneux, Phys. Rev. 30, 1893 (1984); (d) T. Erneux and P. Mandel, ibid., 30, 1902 (1984); (e) N. B. Abraham, Y.