Negative-coupling resonances in pump-coupled...

Negative-coupling resonances inpump-coupled lasers

Thomas W. Carr*, Michael L. Taylor* & Ira B. Schwartz†

[email protected]

*Department of Mathematics

Southern Methodist University

Dallas, Texas

† Nonlinear Dynamical Systems Section

Naval Research Laboratory

Washington, DC

T.W. Carr, Negative Coupling Resonances – p.1

Abstract

We consider coupled lasers, where the intensity deviations from the steady state, modulate

the pump of the other lasers. Most of our results are for two lasers where the coupling

constants are of opposite sign. This leads to a Hopf bifurcation to periodic output for weak

coupling. As the magnitude of the coupling constants is increased (negatively) we observe

novel amplitude effects such as a weak coupling resonance peak and, strong coupling

subharmonic resonances and chaos. In the weak coupling regime the output is predicted by

a set of slow evolution amplitude equations. Pulsating solutions in the strong coupling limit

are described by discrete map derived from the original model.


Laser Physics

Cavity

MirrorR < 1

MirrorR = 1

Resonant

1958 : Townes & Schawlow. Laser Theory. Received Nobel Prize.

Stimulated Emission ofPopulation Inversion

Amplifying Media

Electric &PolarizationFields

Laser

Losses

E2

E1

E1

Pump Population Inversion

ightmplification by

adiation

LASER

mission oftimulated

1960 : Maiman. First operational laser. Ruby cyrstal as amplifier.

E2

ν = Ε2 − Ε1hBeam


Rate equations with pump coupling

I = |E|2 : Intensity, D : Inversion

dIj

dt= (Dj − 1)Ij

dDj

dt= ε2j [Aj − (1 + Ij)Dj ]

Non-zero steady-state corresponds to CW output.

Dj0 = 1, Ij0 = Aj − 1

Investigate the effects of coupling two lasers through their pump:

Aj = Aj0 + Ij0δk(Ik − Ik0).


Pump-coupled Lasers

Laser 1

Laser 2

δ 1

δ 2


Rate equations with pump coupling

Define new variables for the deviations from the cw state as

Ij = Ij0(1 + yj), Dj = 1 + εj√

Ij0xj , tnew = ε1√

I10told.

The new rate equations are

dy1

dt= x1(1 + y1)

dx1

dt= −y1 − εx1(a1 + by1) + δ2y2, δ2 < 0

dy2

dt= βx2(1 + y2)

dx2

dt= β[−y2 − εβx2(a2 + by2) + δ1y1]

where aj , b and β are dissipation & pump constants.


Hopf, Period-doublings and Chaos

For fixed δ1 while varying δ2:Hopf bifurcation at δ2 = δ2H .

δ1δ2 + ε2[a1a2 + 4α

2a1a2

(a1 + a2)2] = 0

Period doubling → chaos. Saddle-node bifurcations → subharmonic resonances.

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

|δ2|

max

(x1)

HB

PD

PD

SN

PDFig. 3c →

← Fig. 4 ↑Fig. 2 & 3a

↓Fig. 3b


Small coupling: δ2 = O(ε)

Look for slowly-evolving small-amplitude solutions of the form

yj1(t, T ) = Aj(T )eit + c.c.,

Obtain the following slow-evolution equations for the amplitudes:

dA1

dT= −

1

2a1A1 −

1

6i|A1|

2A1 −1

2iδ2A2

dA2

dT= −

1

2a2A2 −

1

6i|A2|

2A2 −1

2iδ1A1 + iαA2

To analyze let Aj(T ) = Rj(T )eiθj(T ) and ψ = θ2 − θ1.


Small coupling resonance

R4

2 = 3∆2

1

∆2

∆1 = (δ1δ2 + a1a2)(a1 + a2)2

a1a2

and ∆2 = 1 +a2δ2

a1δ1

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

|d2|

|x2|:

Inve

rsio

n

(a)

300

300−1

7

−0.4

0.5

t

t

(a1): Intensity

(a2): Intensity


3 Lasers

0

1

2

3

4

5

00.5

11.5

22.5

33.5

0

1

2

3

|d2||x

3|

|x1|

SN

SN

u ns t ab le

HB


Large coupling: δ2 = O(1)

0 5 10 15 20 25 30

−4

−2

0

2

4

6

8

x 1 & y

1

(a)

0 5 10 15 20 25 30−0.5

0

0.5

x 2 & y

2

t

(b)

t0 t2

t1

L1: Matched Asymptotics. Pulse = inner region.

L2: Multiple Scales. Damped oscillations in outer. Kick by pulse in inner.


Map

G(t) = (x1 −1

γ)e−γt +

1

γ+

δ2

α2 + ω2e−γt

·[

(αy2 − ωx2)(eαt cos(ωt) − 1) + (ωy2 + αx2)e

αt sin(ωt)]

∫ P

0

G(t)dt = 0

x2 7→ e−1

2εa2P [x2 cos(ωP ) − y2 sin(ωP )] + δ12G(P )

y2 7→ e−1

2εa2P [x2 sin(ωP ) + y2 cos(ωP )]

x1 7→ −G(P ) +2

3εbG(P )2


Fixed points = periodic solutions

0 1 2 3 4 5 6 70

1

2

3

4

5

|x1|

(a)

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

|x2|

(b)

0 1 2 3 4 5 6 75

6

7

8

9

10

Per

iod

(c)

|δ2|


Summary

Negative coupling = phase shift.For nearly-harmonic solutions.

Equivalent to delay of period/2.

Amplitude resonance for weak coupling.Weakly-nonlinear slow-evolution equations.Predict small-coupling resonance.

Am

plitu

de

Parameter

Am

plitu

de

Parameter

Supercritical Subcritical w/ hysterisis

Am

plitu

de

Parameter

Singular Hopf (supercrit.)

Am

plitu

de

Parameter

"?"−Hopf

Pulsating/Harmonic for strong coupling.Map → fixpoints.Similar to single laser with periodic forcing.Pulsations become larger.

Harmonic solutions become smaller.


Fixed Points

max[x1] = π +

√

3a2

2a1δ1|δ2|,

max[x2] = max[y2] =

√

π22a1

3a2

δ1

|δ2|,

P = 2 max[x1].


L1: Pulsing, L2: Pulsing

Derive subharmonic-Melnikov equations

∫ T

0

−ajx2j + δkxjykdt = 0

Use unperturbed Hamiltonian problem to evaluate integrals.Use matched asymptotics to construct pulsating solutions.

Subharmonic-Melnikov equations conditions become:

(1 +a2δ2

a1δ1)T 2 = 0.

For periodic solutions with T 6= 0, δ2 = δ2S .


L1: Pulsing, L2: Harmonic

Localized solutions.

Derive subharmonic-Melnikov equations

∫ T

0

−ajx2j + δkxjykdt = 0

Poincare’-Lindstedt for L1.Multiple scales for L2.

Reproduce local bifurcation result.


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