Vectorial bistability in a quasi-isotropic laser

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Vectorial bistability in a quasi-isotropic laser

D. Hugon, G. Stéphan

To cite this version:

D. Hugon, G. Stéphan. Vectorial bistability in a quasi-isotropic laser. Journal de Physique, 1986, 47

(2), pp.153-156. �10.1051/jphys:01986004702015300�. �jpa-00210190�

153

Vectorial bistability ^{in a} quasi-isotropic ^laser.

D. Hugon* ^{and G.} Stéphan**

Université de Rennes, Laboratoire de Spectroscopie+, ^{Avenue du Général}

Leclerc, ^{35042 Rennes} Cedex, France

(Reçu ^{Ze 29 août} 1985, aaaepté ^{Ze 25 novembre 1985)}

Résumé.2014 Nous montrons théoriquement ^{et ex-} périmentalement que le domaine de bistabi- lité vectorielle observable lors du bascu- lement de polarisation ^{dans un laser} quasi- isotrope ^diminue quand l’intensité de la lumière augmente.

Abstract.2014 The domain of vectorial bistabi- lity observable in the polarization flip ^of

a quasi-isotropic ^{laser is} theoretically

and experimentally ^shown to decrease when

light intensity ^increases.

LE JOURNAL DE PHYSIQUE

J. Physique ⁴⁷ (1986) ^{153-156 FÉVRIER} 1986, ^PA

Classification Physics ^Abstracts

42.55 B - 42.60 H

The polarization ^of light ^{in a la-}

ser can generally ^{be defined} by ^two eigen-

vectors. Quite ^often these vectors do not

depend ^on the laser frequency ^{and their ei-} genvalues ^{are so} different from each other that the light polarization ^remains always the same. This is the case of lasers con-

taining ^{Brewster windows. But in the case} of a quasi-isotropic laser the eigenvectors

may be frequency-dependent ^{and vary within} the emission line. And even if they ^didn’t their associated eigenvalues might ^{show a}

variation introducing ^a polarization flip.

We have recently produced [1] ^{such an ef-} fect using ^{a 3.39} um single ^{mode He-Ne la-}

ser. In this experiment ^the amplifying ^tube

was closed with quasi-perpendicular ^windows

at 86° to the laser propagation ^{axis z} (Fig. 1). Therefore there was a slight pha-

se and loss anisotropy ^{and the} preferred polarization of the tube was in the windows incidence plane yz. However this anisotropy

could be neglected ^when compared ^{to that} induced by ^the optical feedback of the de- tector system. ^The polarizer ^{had its axis} aligned ^{with the} cavity y axis. Therefore the eigenvectors ^{of the} system ^{were unchan-} ged ^when optical ^{feedback was taken into}

account. This effect can be accounted for

Fig. ^1.- Experimental arrangement. ^{1 : mir-}

ror Ra ^{(plane mirror) ; 2 : mirror} Rb ^(con-

cave mirror) ; ^{3 :} polarizer ; 4 : detector (InAs) ; ^{5 : LiF} lens ; 6 : tube window ; 7 : piezo-electric ceramic ; ^{8 :} magnetic shielding ; ^{9 :} diaphragm. ^{d is the} geome- trical length of the laser and L is the _op- tical distance between the output ^mirror Rb

and the detector. In this experiment ^the optical feedback is provided by ^the light diffused by ^{the detector. Numerical data}

are the same as in ref. 1.

theoretically by ^an anisotropic reflectivi- ty ^{of the} exit mirror Rb : Rx = Rb ^{on the x}

axis and Ry ⁼ Rb+ ^E exp (iO) on ^the y axis.

Rb ^{is the isotropic} reflectivity ^coeffi- cient of the mirror and E and (b represent the feedback amplitude ^and phase. ^Because

E « Rb ^{one writes} IR, ^{I = Rb + E cos 4). Let} n, ^{be the} angular frequency of the laser when oscillating ^{on the} y axis, ^{L the} opti- cal distance separating ^the exit mirror Rb

from the detector and d the cavity length.

As (0=211yL/c, ^one obtains M ⁼ 2TEL/d within

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004702015300

154

Fig. ^2.- Light intensity of the single ^mode

laser vs. frequency ^detected through ^the polarizer. Because L = d, cos ~ varies

slowly ^{and the} region ^{AB of} bistability of polarization ⁱⁿ clearly ^{seen. In} (a) the

discharge current is i c ^{= 5 mA, in (b)} ic =

7 mA in (c) ic = 9 mA. ^{The plane mirror}

vibrates at 50 Hz and the _exposure time was

0.5 s i.e. each picture ^{contains a} superposition ^{of 25} forward and backward

curves. ^{It is} clearly ^{seen that the domain}

AB decreases as light intensity ^increases

in ,this ^{case. Other} experiments ^{with L} > d reveal the same result, but less clearly.

the spectral range aw

itc/d. This ^means that 0 ^will go through ^L/d cycles ^within this range. jR j will then be alternatively larger ^and smaller than Rx ^{and the emission}

line, when observed through ^the polarizer,

will be sliced _up with dark crenels.

This letter is meant to study ^the hysteresis phenomenon ^{of a} polarization flip : ^the flip ^{does not occur at the same} frequency ^for increasing ^or decreasing ^fre- quencies [2]. This is shown in figure 2 in

a case where L=d and where the origin our 40 is adjusted ^{so that} cos 0

^{0 is} within the line. This can be done by ^either rotating the polarizer ^{around its} y axis or slightly tilting ^the fixed mirror (here the output mirror Rb). ^{One can} then observe that the

hysteresis ^or polarization bistability ^ran-

ge decreases with an increase of the laser

intensity.

In order to describe this effect,

we use Lamb’s homogeneous theory ; ^{this is} justified as, in this experiment, ^{we are} studying ^{the laser} polarization ^{and fre-}

quency which are independent of the loca- tion within the sytem. ^The amplifying ^me- dium is represented by ^{a saturated tensor}

a .

Its components may be written as fol- lows when the field is oscillating ^{on the x}

axis [3]:

In these equations S, Sp ^and Sz ^{are sums}

containing squares of electric dipole ^ma-

trix elements. No ^{is the} positive popula- tion difference between the two transition states. E2 ^{is the intensity of the satura-}

ting field. The line shape ^{is described} by the plasma dispersion ^{function Z(C)} (posi-

tive imaginary part) ^and by the two func- tions II ^and I_I ^{(negative real parts).}

Further details may be found in reference [3]. The losses are spread over the pheno- menological ^Lamb’s medium and are represen- ted by ^the following ^tensor [4]:

p (components q>.and PY ^{1 is the} phase angle and Q (components Qx ^and Q y) ^{the quality}

factor associated with each axis at the

corresponding ^resonance frequency ^{for the} empty cavity. By deriving ^{Lamb’s evolution} equations [5] ^we get :

II is the laser resonance frequency ^{and w}

.

the actual field frequency. 0 ^{and E repre-}

sent the field phase ^and amplitude ^varia-

tions. These equations ^{are valid as} long ^as

~

the medium polarization P ^{is constant over}

the time interval considered. A flip ^time is of the order of the field variation time in the cavity, ^i.e. 10-8 s for our experi-

~

mental setting. P varies with the atomic quantities of the sytems. ^Its characteris- tic time is about 10-’ s. Here we only ^con- sider the _very beginning ^{of the} flip ^and

not the way with which it develops. ^Within this approximation we assume that the field oscillates on the x axis (E ) and we are

x

looking for ^a condition at which the initi- ally inexistent Ey ^{mode (on the y axis) be-}

gins to grow. Equation (3b) will give ^the difference in the speed ^of variation of two

probes : ^one launched along ^the y axis (the rising Ey ^{mode) and the other one superim-}

posed ^{on the} Ex ^lasing field. However befo-

re making ^the calculation we have to look at the actual Q ^{factor seen} by Ex. ^{For this}

purpose let us first examine the definition of the quality ^{factor in the case of a} pas- sive cavity with mirrors characterized by reflectances R. ^and Rb . ^{In this cavity af-}

ter a round trip ^the field becomes :

with 6t ⁼ 2d/c and 9 the usual phase P ^{= -} 2wd/c . Therefore E ⁼ E0 ^{e-yt with}

T 2- _2d (1- Ra Rb ^{e1.). The intensity varies}

with the real part ^{of T :} I = I o e-Zyrt ^and

yr - _c (1-R Rb ^cos q>). Q ^{which is} by ^defi- 2d ^{’ °}

nition Q ⁼ wI/(-dI/dt) ^can then be written

as follows : Q ⁼ wd/(l- RaRb _c ^{cos(P). By ana-}

logy ^and following ^{Lamb’s method} [6], ^we attribute Qx y (n,) ^{to the rising} Ey mode ^such

as : Q. (11 ⁼ _2fl (04/ ^{C (1- R a IR v [ cos 9 xv ) with}

d

[7] Oxy loy ⁺ y ^{(1 +} 0153r ’ 12£ 0 ). ^{Even when}

4P. = T. (isotropic cavity) one can see that

Q ^{is out of} resonance as e pe y ar . x ^{Now by}

writting equation (3b) separately ^{for each}

Ex and Ey ^{f ield} with y y

= (X x,y Ex y

and by subtracting ^their speed of variation :

When dV becomes positive, Ey grows faster than Ex ^{which means that the} polarization

will switch to the _y axis. This equation

contains several different terms each of them representing ^a physical ^mechanism.

When a term is positive ^{it tends to incre-}

ase OV and to accelerate the apparition ^of

the flip. ^{While if it is} negative ^it pre- vents the laser from switching ^{to the} Ey

mode. In order to separate ^{the different} terms, ^we first ignore the saturation ef- fects :

a) Let us consider an empty cavity ^without

any phase anisotropy.

At the frequency 91v for ^which R - Rx ^beco-

mes positive ^an Ex ----> Ey f lip ^{will be ob-}

served. This is illustrated schematically in figure ³ where the curve a shows an in-

Fig. 3.- Scheme showing ^{the four} different mechanisms contributing ^to AV. Curves a,b, c,d, and a’, b’, c’, d’ are related respec-

tively to the E -->

E ^and E ^{--+ E flips.}

(a) (and (a)) describe the variation with the laser length ^{of the} difference R - R

(and Rx - Ry ) ^{in the region of} changing

sign. In the preceding experiment R - R =

t cos

(2fl L/c) ^{and the sign} variation was

primarily responsible for the polarization flip. (b) (and (b’)) are obtained when

and 1- 0153 1 (n ) I - !- 0153 1 (n ) I > 0.

These curves escribe

Ry y Rx 12 ) x ⁺ 1° or.

I (ay h I (x I en" ) I / &

^{and show the}

role of the inear _gain originating ^from

the frequency difference between the two ween the two fields E and E for a same

x y

length ^{d in a} laser having ^a linear phase anisotropy. (c) (and (c’)) show the influ-

ence of the saturation on

Q.v ^(and Qy x ) :

curves (b) and (b’) are lowered together.

(d) (and(d’)) show the influence of the sa-

turation dichroism on Qt.1 - Qt.1 which rise

y x

curves (c) and (c’).

A and B are the points where polarization flips ^{occur. It is seen that the} domain of

bistability AB depends strongly ^{on the} pha-

se anisotropy ^and that it decreases when

the light intensity increases : in this ca-

se the (d) and (d’) curves rise together

and the points ^{A and B draw nearer.}

156

crease in Ry- Rx ^{with an} increasing ^fre-

quency. ^In the same way and E -+ E flip will occur while decreasing the frequency

when Rx- Ry becomes ^{positive. Obviously}

the two flips ^{occur at the same} frequency.

b) If we consider now a cavity ^with a phase anisotropy, ^{the resonance} frequencies 12 y

and 11 y ^{for the two modes} Ex ^and Ey ^are

different although ^the geometrical length of the laser is the same. One has :

°03B1 is the linear part ^{of a. Here} it is the total quantity (gain-losses) ^{which must be} compared ^{for each} pair ^of frequencies. ^We

have chosen the case

|°03B11i(fl y I > 0a1 (a ) x I

to draw the curve b in figure : ^X

to draw the curve b in figure ^: being positive, ^{the term} O(xl (fly Iocr.) (ill[ ) ac-

celerates the appearance o ^{the phenomenon}

which occurs now at a lower frequency. ^In the same manner curve b’ is obtained from

curve a’ by adding ^{the term}

Here we assume that n n ^x y ^{so that no}

mode hopping ^can appear.

c) The saturation phenomenon first affect the quality ^factor ax,. ^Because 8 ^{is not}

in general ^a multiple ^of 2n it decreases

Qxy on ^{the y axis :}

Qx y Qy ^{(we have} Q. = ’*d/ (1 - Ra _c IR I) ^at

resonance). Therefore it slows down the de- velopment ^{of a} By ^{mode : on figure 3 this}

effect corresponds to ^an additional negati-

ve term to the curve b. This results in

curve c. In the same way it also lowers the b’ curve giving ^{curve c’.}

d) Then the saturation phenomenon ^induces

an anisotropy ^{of the} gain. When Ll. = 12 y ^one

may write :

This is a positive ^{term as the} gain ^{is lar-}

ger ^on the _y axis due to Zeeman coherences of the atomic sublevels. This last effect therefore accelerates the development ^{of a}

E mode. It then rises curve c on figure ³

and results in curve d which crosses the w

axis at point ^{B where the} flip ^{occurs. The}

same effect applied ^{to the situation where}

the Ey ^mode oscillates and frequency ^decre-

ases lead to an increased gain ^{on the x} axis which accelerates also the advent of the flip (point ^{A on} Fig. 3).

It appears that the saturation induced ani- sotropy ^{of the} gas is responsible ^{for the} polarization bistability domain in our ex-

periment : ^{its real} part (induced ^birefrin- gence) ^{tends to increase the} stability ^{of a} mode by decreasing ^the Q ^{factor of the} other while its imaginary part (induced ^di- chroism of amplification) ^{tends on the con-} trary ^{to decrease its} stability by ^increa- sing ^the gain ^{on the} other axis. This ani- sotropy depends strongly ^on the structure of the lasing ^{levels. Now it can be seen} that, depending ^{on the} respective strength of the opposite ^{effects c and} d, the bista-

bility range will be a complicated ^function of the light intensity. ^{However when the} intensity ^{increases the saturated index} tends to 1 and the c effect on the quality

factor decreases while the d effect or di- chroism increases. As a results the polari- zation flip ^will occur sooner and the bis- tability range decreases according ^{to the} experimental results shown in figure ^2.

REFERENCES

[1] Stephan ^{G. and} Hugon D., Phys. ^Rev.

Lett. 55, (1985) 703

[2] Culshaw W. and Kannelaud J., Phys. ^Rev.

141, (1966) 237

[3] Stephan G., Le Naour R. and Le Floch A., Phys. ^Rev. A17, (1978) 733

[4] Hugon D., 3rd cycle thesis, ^Rennes (1982).

[5] Sargent III M., Scully ^{M.L. and Lamb} Jr.

W.E., ^Laser Physics, (Addison-Wesley, London) 1974, P. 101 eq. 14 and 15.

[6] ^Lamb’s formulation has the advantage ^to separate clearly gain and losses. An alternative method consists in writing :

$$ Q-1xy ^{= c/d03C9} {1 - Ra |Ry| ^{cos 03B8xy}

$$ exp(-2 kiy d)}

with the gain exp $$ (-2 kiyd) included in

Qxy. ^This method leads to the same con-

clusions.

[7] Supercripts ^{r and i} stand respectively

vely for "real part of" and "imaginary

part ^of".