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Self-similar pulses in coherent linear

amplifiers

Soodeh Haghgoo and Sergey A. Ponomarenko∗

Department of Electrical and Computer Engineering, Dalhousie University,

Halifax, NS, B3J 2X4, Canada

∗serpo@dal.ca

Abstract:

existing in homogeneously broadened amplifying linear media in a vicinity

of an optical resonance. We demonstrate numerically that the discovered

pulses serve as universal self-similar asymptotics of any near-resonant short

pulses with sharp leading edges, propagating in coherent linear amplifiers.

We show that broadening of any low-intensity seed pulse in the amplifier

has a diffusive nature: Asymptotically the pulse width growth is governed

by the simple diffusion law. We also compare the energy gain factors of

short and long self-similar pulses supported by such media.

We discover and analytically describe self-similar pulses

© 2011 Optical Society of America

OCIS codes: (320.0320) Ultrafast optics; (320.5550) Pulses; (320.5540) Pulse shaping;

(260.0260) Physical optics; (260.2030) Dispersion.

References and links

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in nonlinear optical fibers,” J. Opt. Soc. Am. B 10, 1185–1190 (1993).

7. K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave break-

ing in erbium-doped fiber amplifiers,” Opt. Lett. 21, 68–70 (1996).

8. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, “Self-similar propagation and

amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).

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Phys. Rev. A 82, 021805(R) (2010).

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Rev. A 81, 051801(R) (2010).

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16. K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,”

Phys. Rev. Lett. 90, 203902 (2003).

17. S. V. Manakov, “Propagation of an ultrashort optical pulse in a two-level laser amplifier,” Sov. Phys. JETP 56,

37–44 (1982).

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Lett. 50, 495–498 (1983).

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59, 703–709 (1984).

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A 82, 051801(R) (2010).

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22. P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1985).

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2058–2060 (2010).

24. A small deviation from the universal asymptotics in the pulse tails can be explained by limited accuracy of our

sharp leading edge approximation.

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1.Introduction

Only not too long ago did the optical community realize [1] that self-similarity is quite a ubiq-

uitous feature of optical systems. The phenomena as diverse as optical pulse evolution in Hall

gratings [2], stimulated Raman scattering [3], formation of self-written waveguides [4], and

fractal formation in nonlinear media [5] exhibit self-similarity in one form or another. Recently,

long-term self-similar evolution of pulses in nonlinear fiber amplifiers [6–8], in passive fibers

of lasers [9] has received much attention due to its fundamental interest and potential applica-

tions. Lately, fiber lasers with self-similar evolution in the amplifier and soliton evolution in the

anomalous dispersion segments [10] as well as all-normal-dispersion lasers working in a self-

similar light propagation regime [11] were proposed and experimentally realized. Self-similar

dynamics of beams in nonlinear waveguide amplifiers and in conservative nonlinear media have

also been explored [12–16].

Self-similar evolution of pulses and beams in resonant media has also been explored. In par-

ticular, universal quasi-self-similar asymptotics of ultrashort light propagation in coherent non-

degenerate and degenerate nonlinear amplifiers was examined in Refs. [17] and [18], respec-

tively. Also, self-similarity in superfluorescence in homogeneously broadened resonant media

was explored as well [19]. In addition to the early pioneering work [17–19], however, some

recent studies [15,20] show that a wealth of self-similar regimes exists in such media. In partic-

ular, self-similar beams can be generated in cubic-quintic nonlinear media doped with resonant

impurities in the limit of a large detuning from the impurity resonance [15]. By the same token,

we have shown elsewhere [20] that in resonant nonlinear absorbers, self-similar optical kinks

are formed as intermediate asymptotics of any incident pulse with a long tail in the trailing

edge. In this context, it is instructive to explore the possibility of self-similar pulse formation

in resonant linear media. At first glance, the very proximity to optical resonance(s), coupled

with the system linearity, appears to preclude self-similarity of a sufficiently short pulse: Strong

dispersion at resonance(s) would, in general, seem to cause severe pulse reshaping. One would

then also wonder whether self-similar pulses in such media, if any, would be universal asymp-

totics of very weak seed pulses. The affirmative answer to the last question would augur well

for the experimental realization of such similaritons.

In this paper, we demonstrate analytically that self-similar optical pulses, albeit of a highly

asymmetric shape, can indeed propagate in resonant linear amplifiers. Such an asymmetric self-

similar shape is a manifestation of dynamic balance between linear amplification and phase

relaxation processes in resonant propagation of short pulses in the absence of inhomogeneous

broadening and host medium dispersion. We further show that a low-intensity seed pulse of any

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profile with a sharp leading edge evolves into a self-similar one upon propagation inside the

amplifier. The short pulse broadening has a universal diffusive character such that the rms width

grows as a square root of the propagation distance. Thus, we demonstrate, both analytically and

with numerical simulations that the discovered self-similar pulses are universal intermediate

asymptotics in resonant coherent amplifiers. The intermediate character of the asymptotics is

imposed by the system linearity: As long as the pulse area will have grown enough, our linear

approximation surely breaks down; sufficiently small initial pulse areas and/or short enough

amplifier lengths are required for the linear approximation to hold over the entire amplifier

length.

2.Physical model and mathematical preliminaries

We model a resonant medium as a collection of two-level atoms with the resonance frequency

ω0, thereby limiting our consideration to the case of one internal resonance. We assume here-

after that the pulse spectrum is mainly affected by homogeneous broadening, implying that

γ⊥?δ, γ⊥and δ being transverse (dipole moment) relaxation rate and a characteristic spectral

width of inhomogeneous broadening, respectively. Under these conditions, the evolution of a

pulse with the carrier frequency ω in the medium is governed by a reduced wave equation,

∂ζΩ = iκσ; (1)

subject to the slowly-varying envelope approximation (SVEA):

∂ζΩ ? kΩ,

∂τΩ ? ωΩ.

(2)

Here Ω = 2degE/¯ h is the Rabi frequency associated with the pulse amplitude E, N is an atom

density, degis a dipole matrix element between the ground and excited states of any atom, la-

beled with the indices g and e, respectively;κ =ωN|deg|2/cε0¯ h is a coupling constant, k=ω/c,

and Eq. (1) is written in terms of the transformed coordinate and time, ζ = z and τ = t −z/c.

The dipole moment matrix element σ and one-atom inversion w obey the Bloch equations [21]

∂τσ = −(γ⊥+iΔ)σ −iΩw,

(3)

and

∂τw = −γ?(w−weq)−i

2(Ω∗σ −Ωσ∗).

(4)

In Eqs. (3) and (4) γ?is a longitudinal relaxation rates associated with one-atom inversion

damping, Δ = ω −ω0is a detuning from the resonance and weqis a value of the one-atom

inversion in the absence of the pulse (equilibrium).

In the low-intensity limit, the atomic population is hardly affected by the pulse such that the

one-atom inversion is approximately given by its equilibrium value,

w ? weq= ±1,

(5)

where the upper/lower sign corresponds to amplifier/absorber case: All atoms remain in the up-

per/lowerlevel.Physically,thelinearamplificationregimeofaweakprobepulsecanberealized

using a strong pump in a three-level configuration, standard of laser systems, see e.g. [22]. The

latter is illustrated schematically in Fig. 1 where the pump rate P and the upper level relaxation

rate must be large enough, P ? γ⊥and Γ ? γ⊥, to achieve population inversion between levels

“e” and “g”.

Mathematically, the approximation (5) implies linearization of the dipole moment evolution

equation viz.,

∂τσ = −(γ⊥+iΔ)σ ∓iΩ.

(6)

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P

1

?

eq

w

?

?

e

g

Fig. 1. Schematics of a pump-probe three-level system modeling coherent linear amplifier.

The resonant transition takes place between levels e and g.

In the following, we distinguish two limiting cases: “long” pulses, tp? T⊥and “short” ones,

tp≤ T⊥–in the case tp? T⊥, the pulses may be called “ultrashort”–where T⊥= γ−1

characteristic pulse width.

Long pulses. In this case, the atomic variables can be adiabatically eliminated–using Eq. (6)

and equating σ to its quasi-steady-state value with respect to Ω–which will result in the pulse

evolution equation in the form

∂ζΩ = ±

Here we introduced an inverse Beer’s gain/absorption length α and an overall phase accumula-

tion rate β by the expressions,

⊥and tpis a

?α+iβ

2

?

Ω.

(7)

α =

2κγ⊥

γ2

⊥+Δ2,

β =

2κΔ

γ2

⊥+Δ2.

(8)

It follows at once from Eq. (7) that for sufficiently long pulses, any pulse grows/decays expo-

nentially in such a medium, maintaining its overall shape,

Ω(τ,ζ) = Ω0(τ)e±(α+iβ)ζ/2,

(9)

where Ω0(t) describes an initial pulse profile, and Eq. (9) is well-known Beer’s amplifica-

tion/absorption law.

Short pulses. For simplicity, we consider pulses exactly on resonance with the atomic tran-

sition, Δ = 0; the pulse field and dipole moment evolution equations can then be written as

∂ζΩ =i

2γ⊥α0σ,

(10)

and

∂τσ = −γ⊥σ ∓iΩ.

(11)

where α0= 2κ/γ⊥. Our treatment to this point is equally applicable to amplifying and absorb-

ing media. Hereafter we focus on short pulse propagation in amplifiers.

3. Short self-similar pulses

The inspection of Eqs. (10) and (11) reveals that the electric field of a self-similar pulse and

atomic dipole moment profiles ought to be sought in the form

Ω(ζ,τ) = γ⊥θ(τ)Ω[γ⊥τF(ζ)]e−γ⊥τ,

(12)

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and

σ(ζ,τ) = θ(τ)G(ζ)σ[γ⊥τF(ζ)]e−γ⊥τ.

(13)

Here θ(τ) is a unit step function describing a sharp leading edge of the pulse, F(ζ) and

G(ζ) are arbitrary at the moment and Ω and σ are dimensionless functions. Substituting the

Ansatz (12) and (13) into Eqs. (10) and (11), we can show that self-similarity is sustained

provided that

F(ζ) = α0ζ +T⊥/tp;

Further, the dimensionless pulse envelope Ω in the amplifying medium satisfies the equation,

G(ζ) = 1/F(ζ).

(14)

ηΩ??+Ω?−Ω/2 = 0,

(15)

where we introduced a similarity variable η by

η = γ⊥τ(α0ζ +T⊥/tp).

(16)

and the prime denotes a derivative with respect to η.

Analytically solving Eq. (15), we can obtain a self-similar pulse envelope in a linear ampli-

fier. The overall pulse profile can then be represented as

Ω(η,τ) ∝ γ⊥θ(τ)1F1(1/2,1,−2?2η)exp(?2η −γ⊥τ),

where1F1(a,c,x) is a confluent hypergeometric function, and we dropped an arbitrary (small)

initial pulse amplitude. Eq. (17) can be expressed in a more compact form as

Ω(η,τ) ∝ γ⊥θ(τ)I0(?2η)exp(−γ⊥τ),

where I0(x) is a modified Bessel function of zero order. We note that for sufficiently long

propagation distances, α0ζ ? T⊥/tp, the self-similar pulse profile no longer depends on tp,

yielding a universal self-similar profile

(17)

(18)

Ω(τ,ζ) ∝ γ⊥θ(τ)I0(2

?

κζτ)exp(−γ⊥τ).

(19)

It can be inferred from the analysis of Eq. (18) that the pulse evolution is governed by a

synergy of three factors: pulse shape asymmetry, coherent gain and dipole phase relaxation.

In the absence of nonlinearity, the self-similarity arises as a consequence of dynamic balance

between coherent gain and linear damping; the sharp leading edge of the pulse profile ensures

the balance in the absence of bulk medium dispersion and inhomogeneous broadening. Thus,

the asymmetry of a seed pulse shape appears to be the only requirement for self-similarity in

the studied linear system to emerge.

We stress here that in the linear limit, damping of the trailing edge of the pulse by the linear

relaxation processes allows for the finite energy self-similar pulse formation. The situation here

is drastically different from quasi-self-similarity emerging in the nonlinear amplification of

ultrashort pulses–the term should be understood in the sense defined in Sec. 2–studied in [17].

In the latter case, linear damping is negligible and the nonlinearity promotes the emergence of

finite-energy pulses in the amplifying medium. We notice also that the discovered self-similar

pulses have no chirp–since dispersion plays no role here–which sets them apart from more

familiar parabolic pulses in nonlinear fiber amplifiers. The latter require a linear chirp to prevent

wavebreaking[6,8].Wealsonotethatowningtoadifferentphysicalnature,oursimilaritonsare

markedly different from recently discovered quasi-parabolic pulses in nonlinear amplifiers [23].

While the former are chirp-free self-similar pulses, the latter are phase-modulated steady-state

pulses moving with the speed of light.

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Fig. 2. Normalized intensity of a short self-similar pulse as a function of dimensionless

time T = τ/T⊥and propagation distance Z = α0ζ. The pulse intensity is normalized to its

peak value at Z = 0. The initial pulse width is chosen to be tp= T⊥.

We now proceed to describing the properties of new self-similar pulses. In Fig. 2, we dis-

play the self-similar pulse profile evolution as a function of the dimensionless time T = τ/T⊥

for several values of the dimensionless propagation distance Z = α0ζ. The self-similar char-

acter of the pulse dynamics is clearly discernable in the figure. To demonstrate the universal

nature of the discovered self-similar regime, we numerically simulate the evolution of a generic

asymmetric Gaussian pulse, Ω1(t,0) ∝ θ(t)exp(−t2/t2

file with the self-similar asymptotics. The results are presented in Figure 3. To ensure the two

pulses are sufficiently different in the source plane, we take the Gaussian pulse to be half as

long as the self-similar one at Z = 0: tp= T⊥/2. In the inset to the figure, we compare short-

distance pulse dynamics of the two pulses. We see in the figure that although the Gaussian

pulse profile deviates from the self-similar asymptotics over short distances–at least over first

few Beer’s amplification lengths as is seen in the inset–it quickly converges to the universal

asymptotics over longer distances. It then is seen to coincide with the self-similar asymptotics

profile to within numerical round-off errors [24]. We obtained qualitatively similar results for

hyperbolic secantandexponential profileswithcutoffleadingedges:Ω2(t,0)∝θ(t)sech(t/tp),

and Ω3(t,0) ∝ θ(t)exp(−t/tp).

To reenforce the message, we examine the rms width– defined as ΔT =

measured in the units of T⊥–of the universal self-similar asymptotics on pulse propagation in

the amplifier. The averaging is taken over the pulse intensity distribution, for instance,

p), in the amplifier and compare its pro-

?

?T2?−?T?2and

?T2(Z)? ≡

?∞

0dTT2|E(T,Z)|2

?∞

0dT|E(T,Z)|2.

(20)

With the help of the asymptotic expansion of I0(x) [25], an analytical expression for the rms

width can be derived and presented in an exceptionally simple form

ΔT(Z) ?

√3Z/2.

(21)

In other words, the pulse rms width grows with the distance in a diffusive manner with the

effective diffusion coefficient equal to 3αT2

⊥/8 (in original units). We then evaluate and display

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Fig. 3. Normalized intensity of a short Gaussian (solid) and self-similar (dashed) pulses as

functions of dimensionless time T = τ/T⊥and propagation distance Z = α0ζ. The pulse

intensities are normalized to their peak values at Z = 0. The initial pulse width is chosen

to be tp= T⊥/2 and tp= T⊥for Gaussian and self-similar pulses, respectively. The inset

shows pulse dynamics for short propagation distances.

Fig. 4. Average widths of Gaussian, secant hyperbolic and exponential pulses as func-

tions of the dimensionless propagation distance Z =α0ζ. The self-similar asymptotic pulse

width dependence on the propagation distance is shown as the solid curve.

the behavior of asymmetric Gaussian, hyperbolic secant, and exponential pulse widths in Fig. 4.

The self-similar asymptotic pulse width is drawn in a solid curve. In the inset to the figure, we

exhibit the pulse width dynamics over a short range of propagation distances. We can conclude

from the figure that although the width of an arbitrarily shaped seed pulse initially deviates

from the self-similar pulse width, the former asymptotically tends to the latter over a long

enough propagation distance, thereby underscoring the universal character of the discovered

self-similar asymptotics.

Next, we observe that the applicability of SVEA is not, in general, guaranteed for pulses with

sharp fronts. Hence, the presence of a step function has to be physically justified as follows. A

practical realization of an ideal sawtooth-like pulse involves a finite switching time tswdescrib-

ing the fast rise of its leading edge. Thus for a short pulse, tp∼ T⊥, the pulse duration (rise time

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Fig. 5. Energy gain factor G(Z) for a short (solid) and long (dashed) self-similar pulse as a

function of the dimensionless propagation distance Z = α0ζ.

of the pulse front edge) has to be much shorter than the pulse width, yet much longer than an

optical cycle for the SVEA–see Eq. (2)–to hold:

ω−1?tsw?tp∼ T⊥.

(22)

The complimentary conditions (22) can be realized in a laboratory for picosecond pulses in

dilute atomic vapors, say, for which, typically T⊥∼ 1÷10 ps [21] by choosing, for example,

tsw∼ 10÷100 fs. Mathematically, the leading front step function can then be approximated,

for instance, as

θ(τ) ? [1+tanh(τ/tsw)]/2,

with the excellent approximation attainable for tsw= 0.01T⊥.

Finally, we exhibit in Fig. 5 a short-pulse energy gain factor,

?

for the novel self-similar pulses as a function of the propagation distance. The exponential gain

factor for long pulses,

G0(ζ) = exp(α0ζ),

(23)

G(ζ) =

dτ|E(ζ,τ)|2/

?

dτ|E(0,τ)|2,

(24)

(25)

is presented for comparison as well. On comparing the two, we conclude that for sufficiently

long distances, long pulses are amplified much more efficiently than are short ones. This is

because short pulses have very broad energy spectra with large fractions of their energies stored

in the pulse tails. The latter lie well outside of the medium gain spectrum and are then not

efficiently amplified. Narrow spectra of long pulses, on the other hand, fall entirely within the

medium gain spectrum, which results in strong amplification. These qualitative conclusions are

bourn out by the asymptotic analysis yielding the following universal long-term gain behavior

for asymmetric short pulses

G∞(ζ) ∝

eα0ζ

?α0ζ.

(26)

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Hence, comparing Eqs. (26) and (25), we see that the long-pulse gain dwarfs the short-pulse

one in the long-term limit.

In summary, we have discovered a self-similar regime of short pulse propagation in linear

amplifiers in the vicinity of an optical resonance. The novel self-similar pulses have sharp

leading front, resulting in a highly asymmetric sawtooth-like pulse profile. We have shown that

the new pulses serve as intermediate universal asymptotics for any asymmetrically shaped pulse

propagation in resonant amplifiers in the linear regime. We note that our results hold true in the

absence of inhomogeneous broadening. It will be instructive to determine the influence of the

latter on the emergence of universal self-similar asymptotics in the system. This topic will be

addressed in detail in a forthcoming publication.

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