spatiotemporal polarization dynamics in a transverse ... · chaos,cryptographyandcommunications 3...

Eur. Phys. J. Special Topics© EDP Sciences, Springer-Verlag 2014DOI: 10.1140/epjst/e2014-02124-0

THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICS

Regular Article

Spatiotemporal polarization dynamics in atransverse multimode CO2 laser with opticalfeedback

R. Meucci1, K. Al Naimee1,2, M. Ciszak1, S. De Nicola3,4, S.F. Abdalah1,5 andF.T. Arecchi1,6

1 Istituto Nazionale di Ottica-CNR, Firenze, Italy2 Department of Physics, College of Science, University of Baghdad, Baghdad, Iraq3 CNR-SPIN Complesso Universitario di M.S. Angelo, Napoli, Italy4 INFN Sez .di Napoli, Napoli, Italy5 High Institute of Telecommunications and Post, Al Salihiya, Baghdad, Iraq6 Universita di Firenze, Italy

Received 26 February 2014 / Received in final form 26 February 2014Published online xx March 2014

Abstract. We investigate the polarization dynamics in a quasi-isotropicCO2 laser emitting on the TEM

∗01 mode subjected to an optical

feedback. We observe a complex dynamics in which spatial mode andpolarization competition are involved. The observed dynamics is wellreproduced by a model that discriminates between the intrinsic asym-metry due to the kinetic coupling of molecules with different angu-lar momenta and the anisotropy induced by the polarization feedback.We observe various dynamical regimes including chaotic dynamics andshow that feedback changes these states from regular to chaotic andvice versa. Finally, the possible applications to polarization coding arediscussed.

1 Introduction

Laser dynamics in quasi-isotropic gas lasers, where there exists a competition betweenintrinsic anisotropies, due to the kinetic coupling of molecules with different angularmomenta and residual cavity and detuning anisotropies (extrinsic anisotropies) is atopic of particular interest [1–4]. These phenomena were studied on lasers emitting onthe fundamental transverse mode (TEM00 mode); however, allowing the laser emis-sion on higher order transverse modes makes it possible to investigate the interplaybetween spatial and polarization effects [5,6]. Particular importance is played by theannular mode, more precisely by the TEM01∗ mode. This mode can be considered asthe superposition of two Laguerre Gauss modes or the superposition of two HermiteGauss modes TEM01 and TEM10.In a previous study [7], we observed that, if the polarization state is analyzed

along the H-V axis, two configurations can exist. In particular, depending on thecavity detuning, the annular pattern consists of a TEM01 mode polarized along agiven eigendirection (H or V) and a TEM01 mode polarized along the orthogonal

2 The European Physical Journal Special Topics

Fig. 1. Experimental setup of the CO2 laser with optical feedback: M1 total reflectingmirror; OC output coupling mirror mounted on piezo-translator; LP linear polarizer; FMbeam splitter 50% used as feedback mirror; M2 mirror; L lens; T thermo camera or fastdetector HgCdTe at room temperature.

eigendirection. We named this configuration “splitted”. When the cavity is set farfrom the atomic resonance the pattern changes to a polarized ring pattern alongthe vertical or the horizontal eigendirection (“homogeneus” configuration). However,in these experiments no spontaneous jumps between these configurations where ob-served, in other terms, no pattern alternation was observed in two eigendirections.Improvements in the cavity symmetry has led the actual configuration to observeinteresting spatiotemporal dynamics with evidence of complex dynamics includingchaotic behaviours in the polarized output intensity. Such a dynamics is also accom-panied by spontaneous jumps between the patterns that indicates the existence ofbistability in the system. We also demonstrate that such a dynamics can be con-trolled or enhanced by a suitable optical feedback.By insertion of a polarized beam splitter, a fraction of one of the polarized radi-

ations is rejected in the optical cavity, allowing a control of the polarization state ofemission. This offers the possibility to use coding schemes where both spatial patternsand their polarizations can be exploited, offering a better performance with respectto the laser emission on the fundamental mode where the only degree of freedom forsecurity is offered by the direction of polarization. Finally, we present a mathematicalmodel that reproduces most of the phenomena observed in the experiments.

2 Experimental setup

The polarization dynamics of a quasi-isotropic CO2 laser controlled by optical feed-back has been investigated by using the experimental setup shown in Fig. 1. It consistsof a quasi-isotropic CO2 laser, with a Fabry-Perot optical resonator made of a totallyreflective mirror M1 and an output coupler mirror (OC) with reflectivity R = 90%,radius of curvature ρ = 5m, mounted on a piezoelectric transducer (PZT) in orderto control the laser detuning. The laser tube (LT) is terminated by an antireflec-tion coated ZnSe window and the totally reflective mirror M1. The cavity length isL = 850mm while the discharge length is about 550mm. The active medium is a mix-ture of 82% He, 13.5% N2, and 4.5% CO2 gases pumped by a dc discharge current.

Chaos, Cryptography and Communications 3

The laser is allowed to operate on the first order transverse mode (TEM∗01 mode) bymeans of an intracavity iris diaphragm.The laser beam is directed toward an infrared camera T (Pyrocam III- pixel size

85× 85µm) after passing through a linear polarizer (LP) at 45 with respect to thebeam direction. The beam transmitted by LP is vertically polarized (VB) and thereflected part is horizontally polarized and it is directed towards a totally reflectingmirror M2 through a feedback mirror. The feedback mirror (FM) is a ZnSe non polar-izing beam splitter (50/50). The reflected beam from FM is re-injected into the lasercavity as an horizontally polarized optical feedback beam (HB), whereas the beamtransmitted by FM is reflected by the mirror M2 towards the infrared camera T.Both HB and VB beams are focused onto different pixel areas of the infrared cameradetector surface by means of a lens L.Together with the polarized spatial patterns on the infrared camera T we recorded

also the temporal dynamics by means of a fast HgCdTe detector at room temperatureinserted in one of the two polarized beams.

3 Experimental results

Setting the discharge current at rp = 1.8 times the threshold value and adjusting thecavity detuning, we see the occurrence of a complex dynamics showing competitionbetween polarization and spatial modes. Figure 2 shows polarized spatial patternsrecorded by the infrared camera in absence of optical feedback (g = 0) for increasingvalues of the detuning δ, normalized to the free spectral range of the optical cavity.Polarized spatial patters recorded close to the laser threshold (rp = 1.1) are

shown in Fig. 3. Now, the spatial patterns tend to align and are quite insensitiveto detuning variations. This means that the optical feedback (g = 0.005) induces asymmetry breaking of the unperturbed annular transverse mode (TEM∗01 mode).

We identified various dynamical regimes occuring in the absence and presence ofoptical feedback. In Fig. 4(a-b) we report the maxima of oscillations xm for varyingδ parameter, at zero and non-zero optical feedback g, respectively. Application offeedback changes the dynamics of the system, mostly consisting of the transitionfrom periodic oscillations to steady state, and from periodic oscillations to chaotic. Wehave used the embedding technique to reconstruct the attractor from the single timeseries. In order to decide if the time series in the figures are periodic, quasiperiodicor chaotic we have used the reconstructions of embedded phase spaces to estimatecorresponding correlation dimensions (according to Ref. [8]). As an example, we showthe times series for a free running laser with periodic, chaotic and quasiperiodicoscillations in the intensity (Fig. 4c-e respectively).Moreover, we have found the bistability in the system. A bifurcation diagram

demonstrating the existence of hysteresis obtained experimentally for g = 0 is shownin Fig. 5a, where the maxima of oscillations xm have been reported. In Fig. 5b thetime series in bistable region (for δ = −0.4) are shown.In the following section we introduce theoretical model of the experiment and

demonstrate numerically that the phenomenology observed in the experiment maybe well reproduced.

4 Theoretical and numerical analysis

The theoretical analysis is based on the theory of the isotropic laser [2,3] where thesingle-longitudinal-mode Maxwell-Bloch equations for a polarized two-level laser inthe rotating and slowly varying amplitude approximations and first order coherences


(a) (b)

(c) (d)

(e) (f)

Fig. 2. Polarized spatial patters recorded by the infrared camera for increasing values ofthe detuning in the absence of optical feedback. In each recorded frame, the upper rightpatterns are the spatial profile of the vertically polarized beam; the lower left patterns arethe spatial profile of the horizontally polarized beam: (a) δ = 0; (b) δ = 0.066; (c) δ = 0.133;(d) δ = 0.6; (e) δ = 0.66; (f) δ = 1.

(a) (b) (c)

Fig. 3. Polarized spatial patters recorded near threshold at rp = 1.1 for different values ofthe detuning: (a) δ = 0; (b) δ = 0.066; (c) δ = 0.53.


Fig. 4. Bifurcation diagrams from the experiment (a) g = 0 and (b) g = 0.005 for rp = 1.8.Reconstructed attractors from the time series running in the absence of optical feedback for(c) δ = −0.1, (d) δ = 0.36 and (e) δ = 0.41 with x1 = power (t) and x2 = power (t− τ) forτ = 0.8µs.

between upper levels are considered. The theory was developed for the simplest caseof a J = 1→ J = 0 transition but it has also been applied successfully to explain thepolarization dynamics on the fundamental transverse mode [9] and on the TEM01∗mode [7] in a higher order transition (J = 19→ J = 20) such as a CO2 laser emittingon the P(20) line.The evolution equation for the electric, matter polarization and population in-

version fields can decomposed in a circularly polarized basis and written for eachcomponent of polarization as

ER = k(PR − ER) + iδER + ia(∆⊥ − 4r2)ER − igER − igELEL = k (PL − EL) + iδEL + ia(∆⊥ − 4r2)EL + igER + igELPR = −γ⊥(PR −DRER − ELC)PL = −γ⊥(PL −DLEL − ERC∗) (1)

DR = −γ‖[DR − rp + 1

2(ERP

∗R + E

∗RPR) +

1

4(ELP

∗L + E

∗LPL)

]

DL = −γ‖[DL − rp + 1

2(ELP

∗L + E

∗LPL) +

1

4(ERP

∗R + E

∗RPR)

]

C = −γcC −γ‖4(E∗LPR + ERP

∗L).


Fig. 5. Bifurcation diagram demonstrating the existence of hysteresis obtained from exper-iment for g = 0. (a) Bifurcation diagram showing the maxima of oscillations. (b) The timeseries in the bistable region are shown for δ = −0.4.

In Eqs. (1) ER, EL are the slowly varying electric fields for right R and left L polar-ization, respectively. They are related to the fields Ex and Ey in the H-V orthogonal

basis by the relations Ex = (ER + EL)/√2 and Ey = i(ER − EL)

/√2.

The matter polarization fields are PR, PL and DR, DL the corresponding popu-lation inversions, C(t) is the first order coherence between the sublevels in the caseof the transition from a state J = 19 to J = 20.The coherence accounts for the induced anisotropies in the laser medium responsi-

ble for the competition between the two polarized modes, rp is the pump normalizedto its threshold value, the parameter k = −c log(R)/4L represents the cavity losseswhose value, according to our resonator specifications, is k = 9.3 106 s−1.The optical feedback caused by the presence in the experimental set up (Fig. 1) of

the feedback mirror FM is modeled in the equations for the electric fields by includingterms proportional to the field and the feedback strength g, with g 1. In the or-thogonal basis H-V the feedback term which couples to the horizontal x polarizationstate takes the simple form gEx.The transverse coordinate x− y are rescaled with respect to the minimum beam

waist w0 = λ√L(ρ− L)/π, ∆⊥ = ∂2x+∂2y is the transverse Laplacian, and r2 = x2+y2

is the square of the radial transverse coordinate to the mirror center.The parameter a = λc/4πγ⊥w20 is the diffraction coefficient with λ = 10.6µm the

laser wavelength, δ represents the cavity detuning of the field modes from the mat-ter transition frequency, i.e. the frequency of the P(20) line which is the only activetransition.In our low-pressure, homogeneously broadening CO2 laser, the polarization decay

is γ⊥ = 4.4× 108 sec−1 and the population inversion decay rate is γ‖ =

1.95×105 sec−1 [9]. The parameter γc is the coherence decay rate whose value ischosen between γ‖ and γ⊥ [9].


Fig. 6. Numerical results for the polarized spatial patterns at a pump value rp = 1.8.(a) δ = 0; (b) δ = 0.5; (c) δ = 0.625; (d) δ = 0.8; (e) δ = 1; (f) δ = 1.25.

From the experimental results we know that just the TEM01 and TEM10 modestake part in the dynamics, and therefore the most general expression for the slowlyvarying electric field operating on these modes can be written as time dependentsuperimpositions of mode functions

ER(r, t) = αR(t)A1(r) + βR(t)A2(r)

EL(r, t) = αL(t)A1(r) + βL(t)A2(r)(2)

In Eq. (2), the mode functions A1(r) = 2xA0(r) and A2(r) = 2yA0(r) with A0(r) =√2/πe−r

2

are the standard orthonormal Gauss-Hermite modes TEM01 and TEM10and the time dependent functions αR(t)βR(t), αL(t)βL(t) describe the slowly varyingtime dependence of the fields.


Fig. 7. Calculated polarized spatial patterns at detuning δ = 0 for increasing values ofthe pump parameter: (a) rp = 1.1; (b) rp = 1.7; (c) rp = 2.1; (d) rp = 2.2; (e) rp = 2.3;(f) rp = 2.4.

The dynamics of the system can be studied according to the above describedmodal expansion. The matter variables PR, PL and DR, DL and the coherence Coperating on these modes can be similarly expanded

PR(L)(r, t) = p1,R(L)(t)A1(r) + p2,R(L)(t)A2(r)

DR(L)(r, t) =

5∑i=0

di,R(L)(t)Bi(r) (3)

C (r, t) =5∑i=0

ci(t)Bi(r)


Fig. 8. Time evolution of the power of the x-polarized spatial pattern (red curve) and y-polarized spatial pattern (blue curve) at zero detuning for increasing values of the pumpparameter: (a) rp = 0.9; (b) rp = 2.1; (c) rp = 2.3; (d) rp = 2.4.

where we have defined the following orthonormal basis to expand the populationinversion and coherence:

B0 =√πA20

B1 =√π(A21 −A22)

B2 =√π(A21 +A

22 −A20)

B3 =√2πA0A1

B4 =√2πA0A2

B5 =√4πA1A2

(4)

Using the modal expansions (2) and (3) into the Maxwell-Bloch system of equa-tions (1) we obtain a set of coupled equations for the electric fields amplitudesαR(t)βR(t), αL(t)βL(t), the amplitudes of the matter variables and coherence. Thecomplete set of equations is reported in the Appendix.The modal expansion method reproduces fairly well the experimentally recorded

spatial pattern. The evolution of the transverse pattern intensity distributions is cal-culated for different values of the optical feedback strength, pump value and cavitydetuning.Figure 6 displays the spatial distribution across the x − y plane of the polarized

patterns calculated for increasing values of the detuning from δ = 0 to δ = 1.25. Thenumerical results are calculated at a normalized pump value rp = 1.8. The upperright and lower left are the horizontal and vertical polarized spatial patterns.


(a) (b)

Fig. 9. Bifurcation diagrams obtained numerically in the cases (a) without optical feedbackfor g = 0 and (b) with optical feedback for g = 0.002. xm mark the maxima of oscillations,with xm = 0 denoting the steady state.

Fig. 10. The global view of the results shown in Fig. 9. Dynamical regimes calculatednumerically as the control parameters are modulated: steady state (black colour), periodicoscillations (gray colour) and aperiodic dynamics (white colour) for (a) g = 0 and (b)g = 0.002.

The numerical simulations of the polarized patterns calculated at resonance forincreasing pump parameter rp are shown in Fig. 7. It can be clearly seen that that thepolarized patterns tend to stay aligned up to a (pump parameter) around rp = 1.7.


Fig. 11. Bifurcation diagrams demonstrating the existence of hysteresis obtained numeri-cally for (a) g = 0 and (b) g = 0.002.

This fact confirms that the observed pattern alternation far from threshold dependson the energy provided by the pump mechanism. This phenomenon was also observedon the fundamental mode in a condition leading to coherence resonance between thetwo allowed polarization configurations [10].The time dependence of the functions αR(t)βR(t), αL(t)βL(t) embodies the ob-

served rich spatio-temporal dynamics. To further analyze the time evolution of thepolarized spatial pattern we have calculated the time behavior of the power of the xand y-polarized pattern. Figure 8 shows the total intensity time dependence of thehorizontal x- and vertical y-polarized spatial pattern for increasing values of the pumpparameter.Power fluctuation can be clearly seen for values of the pump parameter greater

than rp = 2. These fluctuations cause the observed pattern alternation phenomenon(compare Fig. 8b-c and d) and demonstrates the competition between polarized spa-tial modes. The high spikes denotes the transition from below to above thresholdwhen the cavity loss parameter k is changed. Such spikes are the typical ones in classB lasers [11]. For comparison, in Fig. 8a we also plot the condition below thresholdwhere the laser intensity relaxes to the steady state condition with zero intensity.We have shown in Section 3, that laser produces a rich complex dynamics. In

order to understand how the optical feedback changes these dynamical states of the


system, we realize a detailed numerical analysis by scanning various system parame-ters in the presence and in the absence of the feedback. The systematic scanning ofthe parameters (through bifurcation diagrams) allows us to obtain a global image ofthe feedback effect on the system.In Fig. 9 we plot numerically calculated bifurcation diagrams for various values

of the control parameters rp and δ in the absence of optical feedback (Fig. 9a) andin the presence of the optical feedback (Fig. 9b). It can be noticed, by comparingthese two cases, that feedback changes the dynamical regimes of laser by stabilizingor destabilizing the original orbits. Analyzing the phase space of parameters rp andδ (see Fig. 10), we can observe quantitatively the effect of feedback of the system.Here, we consider only one value of optical feedback (assumed to be small with re-spect to the laser intensity), however a similar phenomenology is observed for other(still small) values.In Fig. 11a we show the bifurcation diagrams where parameter rp was increased

and subsequently decreased. The resulting diagrams revealed the existence of hystere-sis in the system. In a certain parameter regions we observe the existence of bistability,between (i) steady and periodic states, (ii) two various amplitude periodic oscillationsand (iii) periodic and chaotic oscillations. Thus numerical results confirm the exis-tence of bistable regimes observed in the experiment. Curiously, numerical simulationsdemonstrate that optical feedback eliminates bistable states (see Fig. 11b).

Conclusions

We have analyzed the evolution of the spatial polarization profiles under the influenceof a polarized optical feedback. Near threshold, a symmetry breaking of the annularpattern weakly dependent on the cavity detuning is observed. Far from threshold,pattern and polarization instabilities are induced leading to an alternation of differ-ent polarized patterns critically dependent on the cavity detuning.We have demonstrated the existence of rich complex dynamics in a system, includ-

ing chaos, quasiperiodicity and bistability, that may be efficiently controlled by anapplication of a weak optical feedback. Moreover we derived a theoretical model thatwell reproduces all the phenomenology observed in the experiment. The exhaustivenumerical simulation of a model delivered an useful information about the effect offeedback in a wide range of system parameters.The possibility to alter the polarization direction in an annular pattern, which is

the usual one in unstable resonators, by means of a weak optical feedback is of crucialimportance in industrial high power applications for cutting processes. Communica-tion systems for near – earth space applications based on a CO2 laser were proposedby J.H. McElory et al. [12]. Considering the interest in secure communication linksbased on chaotic optical carriers [13] we believe that spatial polarization dynamicscould also be usefully employed in telecom based applications where the codificationof information also utilizes the polarization properties of the laser modes. Recently,experimental evidence of polarization chaos in vertical-cavity surface emitting laserswas given by Olejniczak et al. [14] and by Virte et al. [15]. The security enhance-ment of optical chaotic communications based on the polarization properties of theselasers was demonstrated by Xiang et al. [16]. Other communication schemes based onpolarization-rotated optical feedback and polarization-rotated optical injection wereproposed by J. Liu [17].

Work was partly supported by Ente Cassa di Risparmio di Firenze. MC and RM acknowledgeRegione Toscana for financial support.


Appendix: Expansion of the matter variables and coherence

In this appendix we explicitly apply the modal expansion method to the mattervariables and coherence. The time dependence of the modal electric field amplitudefunctions αR(t) βR(t), αL(t) βL(t) is given by the following equations

αR(τ) = k (p1,R − αR) + iδαR − iΩαR − ig2αR − ig

2αL

βR(τ) = k (p2,R − βR) + iδβR − iΩβR − ig2βR − ig

2βL

αL(τ) = k (p1,L − αL) + iδαL − iΩαL + ig2αR + i

g

2αL

βL(τ) = k (p2,L − βL) + iδβL − iΩβL + ig2βR + i

g

2βL

where the dot denotes the time derivative with respect to the dimensionless timeτ = γ⊥t and Ω = 4a. The polarization coefficients pi,R(L)(i = 1, 2) are determined by

p1,R(τ) = −p1,R + 1

2√π(d1,RαR + c1αL + d2,RαR + c2αL + d5,RβR

+ c5βL + d0,RαR + c0αL)

p2,R(τ) = −p2,R + 1

2√π(d2,RβR + c2βL − d1,RβR − c1βL + d5,RαR

+ c5αL + d0,RβR + c0βL)

p1,L(τ) = −p1,L + 1

2√π(d1,LαL + c

∗1αR + d2,LαL + c

∗2αR + d5,LβL

+ c∗5βR + d0,LαL + c∗0αR)

p2,L(τ) = −p2,L + 1

2√π(d5,LαL + c

∗5αR − d1,LβL − c∗1βR + d2,LβL

+ c∗2βR + d0,LβL + c∗0βR) .

The amplitude coefficients of the population inversions are

d0,R(L)(τ) = −γ‖γ⊥d0,R(L) +

γ||γ⊥

√πrp −

γ‖γ⊥

(Re(f0,R(L)

)+1

2Re(f0,L(R)

))

d1,R(L)(τ) = −γ‖γ⊥d1,R(L) −

γ‖γ⊥

(Re(f1,R(L)

)+1

2Re(f1,L(R)

))

d2,R(L)(τ) = −γ‖γ⊥d2,R(L) +

γ‖γ⊥

√πrp −

γ‖γ⊥

(Re(f2,R(L)

)+1

2Re(f2,L(R)

))

d3,R(L)(τ) = −γ‖γ⊥d1,R(L) −

γ‖γ⊥

(Re(f3,R(L)

)+1

2Re(f3,L(R)

))

d4,R(L)(τ) = −γ‖γ⊥d2,R(L) −

γ‖γ⊥

(Re(f4,R(L)

)+1

2Re(f4,L(R)

))

d5,R(L)(τ) = −γ‖γ⊥d2,R(L) −

γ‖γ⊥

(Re(f5,R(L)

)+1

2Re(f5,L(R)

)).


Where we have defined the following functions

f0,R(L) =1

2√π(αR(L)p

∗1,R(L) + βR(L)p

∗2,R(L))

f2,R(L) = f0,R(L)

f1,R(L) =1

2√π(αR(L)p

∗1,R(L) − βR(L)p∗2,R(L))

f3,R(L) = f4,R(L) = 0

f5,R(L) =1

2√π(αR(L)p

∗2,R(L) + βR(L)p

∗1,R(L)).

Finally, the amplitude coefficients of the modal expansion of the coherence are givenby the equations

c0(τ) = − γcγ⊥c0 −

γ‖8√πγ⊥(α∗Lp1,R + αRp

∗1,L + β

∗Lp2,R + βRp

∗2,L)

c1(τ) = − γcγ⊥c1 −


∗1,L + β

∗Lp2,R + βRp

∗2,L)

c2(τ) = − γcγ⊥c2 −


∗1,L + β

∗Lp2,R + βRp

∗2,L)

c3(τ) = − γcγ⊥c3

c4(τ) = − γcγ⊥c4

c5(τ) = − γcγ⊥c5 −


∗2,L + β

∗Lp1,R + βRp

∗1,L).

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