Download Complete energy conversion by autoresonant

Complete energy conversion by
autoresonant three-wave mixing in
nonuniform media
O. Yaakobi∗ , L. Caspani, M. Clerici, F. Vidal, and R. Morandotti
INRS-EMT, University of Quebec, 1650 Boul. Lionel Boulet, Varennes, Quebec, J3X 1S2
Canada
∗ [email protected]
Abstract:
Resonant three-wave interactions appear in many fields of
physics e.g. nonlinear optics, plasma physics, acoustics and hydrodynamics.
A general theory of autoresonant three-wave mixing in a nonuniform
medium is derived analytically and demonstrated numerically. It is shown
that due to the medium nonuniformity, a stable phase-locked evolution
is automatically established. For a weak nonuniformity, the efficiency of
the energy conversion between the interacting waves can reach almost
100%. One of the potential applications of our theory is the design of
highly-efficient optical parametric amplifiers.
© 2013 Optical Society of America
OCIS codes: (190.4223) Nonlinear wave mixing; (190.4410) Nonlinear optics, parametric
processes; (190.4970) Parametric oscillators and amplifiers; (190.4975) Parametric processes;
(350.5500) Propagation; (350.7420) Waves.
References and links
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28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 1623
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1.
Introduction
Autoresonance is a unique property of many driven or mutually interacting oscillatory or wave
systems stemming from their ability to stay in resonance (phase-locking) if the parameters of
the system vary in space and/or time adiabatically [1]. One of the most important advantages of
autoresonant schemes is their ability to transfer energy between the interacting waves with very
high efficiency. In addition to being of fundamental interest, the autoresonance effect enables
numerous exciting applications in many fields of physics and engineering. The autoresonance
mechanism was first employed for relativistic particle acceleration 70 years ago [2, 3], and in
the last 20 years it has found widespread applications to the excitation and control of nonlinear
structures in many other fields (see for instance [1] and references therein). One of the most
significant recent scientific achievements accomplished by implementing autoresonant schemes
was the trapping of cold anti-hydrogen atoms at CERN [4].
In the field of nonlinear optics, adiabatic conversion processes in nonuniform media were
previously studied in various systems involving two-wave interaction [5-13] (in some of these
references the considered interaction is between three waves in the undepleted pump regime,
which is effectively a two-wave mixing process). Other related studies of three-wave interactions in one dimensional nonuniform media were conducted in co-propagating [14-16] and
counter-propagating waves configurations [17-19] as well as in a multidimensional spatiotemporal varying medium [20], addressing specific applications in plasma physics and nonlinear optics. A complete theory of autoresonant Four-Wave Mixing (FWM) in a nonuniform
medium was later developed by one of the authors and co-workers [21], suggesting a new application of Optical Parametric Amplification (OPA) using tapered fibers.
The mathematical description of the aforementioned plasma three-wave interactions included
at least one nonlinear term (named as plasma nonlinear frequency shift) in the evolution equations of the wave amplitudes in addition to the nonlinear coupling terms among the interacting
waves. The existence of nonlinear frequency shift terms with sufficiently large magnitude (satisfying some threshold criteria) appeared to be a requirement for the creation of stable autoresonant states.
In this article, we develop a general theory of autoresonant Three-Wave Mixing (TWM) in
nonuniform media. We show that due to mechanisms different from those previously reported,
it is possible to reach an autoresonant state also in the absence of frequency shift effects. The
theory that is derived in this article is suitable for the description of a large variety of three#179667 - $15.00 USD Received 12 Nov 2012; revised 20 Dec 2012; accepted 21 Dec 2012; published 15 Jan 2013
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wave mixing processes such as stimulated Raman scattering in inhomogeneous plasmas [22],
various acoustic wave interactions in solids [23], hydrodynamic three-wave interactions [24]
etc. Probably, the field in which this study is most beneficial is nonlinear optics. In particular,
new conversion schemes that allow complete pump depletion could be designed based on our
theory, e.g. highly efficient OPAs using a nonuniform χ (2) medium. In order to simplify the interpretation of our theory in this context, we will henceforth adopt the terminology of nonlinear
optics.
2.
Mathematical model
We begin our analysis with the set of equations for slow wave envelopes A j describing the
interaction between three waves at frequencies that satisfy the matching condition ω1 + ω2 = ω3
[25]:
dA j
−i z Δk
= iη j A∗m A3 e 0
dz
z dz
; j, m = 1, 2; j = m,
dA3
+i z Δk
= iη3 A1 A2 e 0
dz
z dz
.
(1)
(2)
In the context of TWM in an optical medium with χ (2) nonlinearity, A j (z) is the envelope of
kj is the
the electric field of the j-th wave such that E j (z,t) = A j (z) exp (i [k j z − ω j t]), where
j-th wave wavevector. The coupling coefficients η j are defined by η j = 2de f f ω 2j / k j c2 where
c is the speed of light in vacuum and de f f is the effective nonlinear susceptibility. We assume
slowly varying wavevectors k j (z) for each of the interacting waves and define the wavevectors
mismatch Δk(z) = k1 + k2 − k3 . For simplicity, in our theory we take into account the medium
nonuniformity only from the presence of the z dependence of Δk(z) in the wavevectors mismatch, i.e. k j in the coupling coefficients in Eqs. (1)-(2) are replaced by an average value k j,avg
along the medium for each j. We now definethe dimensionless
coordinate
ζ = z/l and dimen
−1/2
−1/2
|A3,0 | where l = 1/ (η1 η2 )1/2 |A3,0 | and
A j / η3
sionless complex amplitudes a j = η j
the subscript ”0” denotes initial condition values (note that |a3,0 | ≡ 1). Using these definitions,
Eqs. (1)-(2) are transformed to the dimensionless form:
ζ
da j
−il Δk
= ia∗m a3 e 0
dζ
ζ dζ
; j, m = 1, 2; j = m,
ζ
da3
+il Δk
= ia1 a2 e 0
dζ
ζ dζ
Next, we represent each one of the complex
phase
value B j and real j using the definitions
φ ζ
B3 exp i φ3 + l 0 Δk ζ d ζ , and substitute these
composing each one of the resulting equations into its
following set of equations:
.
(3)
(4)
amplitudes a j by its absolute
a1,2 = B1,2 exp (iφ1,2 ) and a3 =
expressions into Eqs. (3)-(4). Dereal and imaginary parts gives the
dB j
= ε j Bm Bn sin Φ; j, m, n = 1 − 3; j = m = n,
dζ
(5)
dΦ
= lΔk + Q cos Φ,
dζ
(6)
where we definedthe phase mismatch
Φ = φ1 + φ2 − φ3 , the sign coefficients ε1,2 = +1, ε3 = −1
−2
−2
and Q = B1 B2 B3 B−2
1 + B2 − B3 . The set of three real amplitude equations (5) is equivalent
#179667 - $15.00 USD Received 12 Nov 2012; revised 20 Dec 2012; accepted 21 Dec 2012; published 15 Jan 2013
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28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 1625
j
B2
1
3
0.5
B22
0
−6
(Φ/π+1)mod(2)−1
B2
(a)
1
−4
−2
(b)
B21
0
|α|ξ
2
4
6
Autoresonant stage
0
Initial phase−locking stage
−1
−6
−4
−2
0
|α|ξ
2
4
6
Fig. 1. Phase-locked evolution of (a) wave envelopes B2j and (b) phase mismatch Φ versus
normalized distance |α | ξ for a spatial nonuniformity rate α = 0.01. In (a), the idler, signal
and pump are denoted by (1) green, (2) blue and (3) red respectively. The black curves
represent the value of the analytical expressions Eqs. (16)-(17). Note that the curves of the
signal and idler are almost overlapping.
to the following set of two algebraic equations, known as Manley-Rowe relations, and one
differential equation for B1 :
dB1
= B2 B3 sin Φ,
(7)
dζ
B21 − B22 = B21,0 − B22,0 , B21 + B23 = B21,0 + B23,0 .
(8)
For simplicity, we assume in the following that the wavevector mismatch is a linear function
of position and define α as the nonuniformity rate coefficient such that Δk = α (z − z∗ )/l 2 =
α (ζ − ζ∗ )/l where the subscript ∗ denotes the point of perfect wavevector matching Δk = 0.
Then, we define a shifted normalized coordinate ξ = ζ − z∗ /l = ζ − ζ∗ such that the wavevecotors are matched at ξ = 0, and rewrite our system (6)-(7) as:
dB1
= B2 B3 sin Φ,
dξ
(9)
dΦ
= αξ + Q cos Φ.
dξ
(10)
Our goal in the next sections is to study the set of two differential equations (9)-(10) for B1 and
Φ in which the variables B2 and B3 are expressed by Eqs. (8).
In Figs. 1 and 2 we present the numerical solution of Eqs. (9)-(10) as a function of the
normalized distance along propagation |α |ξ . The equations were solved in the interval −6 ≤
|α |ξ ≤ 6 for the initial conditions B1,0 = 0, B2,0 = 0.01, B3,0 = 1 and Φ0 = 0. For the case α =
0.01 which is shown in Fig. 1, we observe that the dynamics can be divided into two stages. The
first stage (|α | ξ < −2) is characterized by an initial phase-locking (Φ ≈ 0), in which the energy
of the pump wave could be considered as constant (undepleted pump regime). Phase-locking is
preserved continuously, and in the second stage (|α | ξ > −2), the pump energy is depleted with
a monotonic increase in the average energy of the signal and idler fields, until almost complete
#179667 - $15.00 USD Received 12 Nov 2012; revised 20 Dec 2012; accepted 21 Dec 2012; published 15 Jan 2013
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j
B2
1
3
0.5
B22
0
−6
(Φ/π+1)mod(2)−1
B2
(a)
1
−4
−2
B21
0
|α|ξ
2
4
6
2
4
6
(b)
0
Initial phase−locking stage
−1
−6
−4
−2
0
|α|ξ
Fig. 2. Evolution of (a) wave envelopes B2j and (b) phase mismatch Φ for large value of
nonuniformity rate α = 1. Due to the large value of α , the initial phase-locking is lost prior
to reaching the point |α | ξ = −2. Note that the curves of the signal and idler (blue and
green, respectively) are almost overlapping.
pump depletion is achieved. In the second case, which is presented in Fig. 2, the nonuniformity
parameter is large (α = 1). In this condition the initial phase-locking is observed again but it
is then lost as |α | ξ → −2. Consequently, the conversion efficiency with respect to the pump
energy is low, and it strongly depends on the position along the propagation. Next, we derive
an analytical theory that explains the features observed in the numerical examples.
3.
3.1.
Initial phase-locking and autoresonant state
Closed-form solutions
We start by showing that the initial dynamics of a TWM process in a nonuniform medium can
be described as a two-wave mixing process. This dynamics is similar to the initial dynamics of a
FWM process in a similar system, as studied in Ref. [21], and will be briefly summarized in this
paragraph for the sake of completeness. In spite of the similarity between TWM and FWM in
the initial stage, there are essential differences between the dynamics of these processes in the
autoresonant stage (|α | ξ > −2). Such differences will be analysed in the rest of this article. Our
study begins at a sufficiently large negative wavevector mismatch |α | ξin . Further, we assume
that the initial signal is small (B2,0 << B3,0 ), and that B1 (z) << B2 in the initial stage, since the
idler is initially zero. Consequently, we neglect the variation of B2 and the depletion of the pump
wave at this stage and focus on the evolution of B1 . Then, Q ≈ B2,0 B3,0 /B1 = B2,0 /B1 , since
according to our definition of variables B3,0 ≡ 1. Therefore, Eqs. (9)-(10) can be approximated
by
B2,0
dΦ
dB1
= B2,0 sin Φ,
= αξ +
cos Φ.
(11)
dξ
dξ
B1
Note that Eq. (11) has the same form as Eq. (11) in Ref. [21]. It was shown there that starting at
a sufficiently large negative value of |α | ξin guarantees phase-locking in which Φ ≈ π for α < 0
and Φ ≈ 0 for α > 0 as seen in Figs. 1 and 2. As the idler wave is excited, the signal amplitude
#179667 - $15.00 USD Received 12 Nov 2012; revised 20 Dec 2012; accepted 21 Dec 2012; published 15 Jan 2013
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28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 1627
cannot be assumed as a constant, and one has to consider the following two-wave system:
dB1
= B2 sin Φ, B22 = B21 + B22,0 ,
dξ
B2 B1
dΦ
cos Φ.
= αξ +
+
dξ
B1 B2
(12)
(13)
The two-wave system of equations (12)-(13) was studied in Ref. [21] (see there Eqs. (13)-(14))
and the analytical solutions of B1 and B2 were found. It was shown that the aforementioned
phase-locking continues, and while approaching the point |α | ξ = −2, the amplitudes of the
idler and the signal become comparable to each other, i.e. B1 ≈ B2 . Beyond this point, one must
take into account the pump depletion in the TWM process, by analysing Eqs. (9)-(10), which
differ from the corresponding FWM equations.
In the region |α | ξ > −2 we assume |cos Φ| ≈ 1, and set B1 /B2 ≈ 1, as discussed above. We
define the autoresonant solution (denoted by the hat symbol) consistently to the requirement
that the left hand side of the Eq. (10) vanishes:
αξ + sQ̂ = 0,
sin Φ̂ =
1 d B̂1
,
B̂1 B̂3 d ξ
(14)
(15)
−2
−2
−2
−2
2
where s ≡ sign(α ) = cos Φ̂ and Q̂ = B̂1 B̂2 B̂3 (B̂−2
1 + B̂2 − B̂3 ) ≈ B̂1 B̂3 (2B̂1 − B̂3 ) = 2B̂3 −
2
2
2
B̂1 /B̂3 . B̂1 can be expressed in
the Manley-Rowe relations in Eqs. (8), from
terms of B̂3 using
which we see that Q̂ ≈ 2B̂3 − B21,0 + B23,0 − B̂23 /B̂3 ≈ 3B̂3 − 1/B̂3 . Substituting the expression
for Q̂ in Eq. (14) results in a quadratic equation for B̂3 . The autoresonant solution for the
pump is chosen as the root of the quadratic equation for which B̂3 > 0. This solution decreases
monotonically with |α |ξ , and is the same for both α > 0 and α < 0,
−|α |ξ + (|α | ξ )2 + 12
, |α |ξ > −2.
B̂3 =
(16)
6
The corresponding expressions for the signal and idler evolution in autoresonance are obtained
using the Manley-Rowe relations Eqs. (8) and are characterized by a monotonic increase as a
function of |α |ξ :
⎡
B̂22 ≈ B̂21 = 1 − ⎣
−|α |ξ +
⎤2
(|α | ξ )2 + 12
⎦ , |α |ξ > −2.
6
(17)
There are interesting differences between the dynamics of the autoresonant FWM process that
was studied in [21] and the TWM process that is considered in this article. Although in both
processes, autoresonant dynamics can be obtained for α with either a positive or a negative
sign, the spatial form of the solutions is the same for either sign in TWM (see Eqs. (16)-(17)),
whereas in FWM it depends on the sign of α . Furthermore, in the FWM process, the spatial
range of the autoresonant stage was found to be limited to a specific range of |α |ξ depending
on the sign of α , after which phase-locking was lost. In contrast, we will show that in the TWM
process, once established, stable autoresonant dynamics continues without any bounds.
This difference between the autoresonant states in TWM and FWM can be understood by
examining the corresponding Q terms in the limit of vanishing pump power B23 → 0 (and B21,2 →
#179667 - $15.00 USD Received 12 Nov 2012; revised 20 Dec 2012; accepted 21 Dec 2012; published 15 Jan 2013
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1). In this limit, for TWM Q → −1/B3 while for FWM Q → const. Recalling that phase-locking
is characterized by the balance between Q and the detuning term |α | ξ , we see that in TWM, as
|α |ξ increases, this balance may be mainatined by a continous monotonic decrease in B3 (and
a suitable increase in B1,2 ). However, in FWM, the constant limit value of Q indicates that at
some point, the necessary balance for preserving phase-locking could not be satisfied, and the
system will depart from the autoresonant state.
3.2.
Stability analysis
We have already discussed the quasi-steady state solution in our system and showed that the
system enters this state automatically in the initial phase locking stage (|α | ξ < −2). We will
show now that for small enough |α | this is followed by a continuous stable autoresonance in
the system. In studying the stability, we assume a solution close to the steady state, i.e. we
write B1 = B̂1 + δ B1 and Φ = Φ̂ + δ Φ where δ B1 , δ Φ << 1. In the following we will use the
approximate trigonometric relations sin Φ = s sin δ Φ, cos Φ ≈ s (derived from sin Φ̂ = 0 and
cos δ Φ ≈ 1). Next, we replace cos Φ by s in Eq. (10) and expand the equation to first order in
δ B1 , yielding
dΦ
= s(|α | ξ + Q̂) + sQ̂ δ B1 ,
(18)
dξ
where Q̂ denotes the derivative of Q with respect to B1 evaluated at B̂1 . Phase-locking is characterized by d Φ̂/d ξ = s(|α | ξ + Q̂) ≈ 0, so we define B̂1 in our system at all stages of evolution
(including the initial phase-locking stage |α | ξ < −2, the autoresonant stage |α | ξ > −2 and
the transition region |α | ξ ≈ −2) by
Then,
Q̂ ≡ − |α | ξ .
(19)
d (δ Φ)
= sQ̂ δ B1 .
dξ
(20)
On the other hand, d (δ B1 ) /d ξ = dB1 /d ξ − d B̂1 /d ξ , which, by using Eq. (9) to the lowest
order in δ B1 , δ Φ and d B̂1 /d ξ = − |α | /Q̂ obtained by differentiation of Eq. (19), yields
|α |
d (δ B1 )
= sD sin δ Φ + ,
dξ
Q̂
(21)
where D = B̂2 B̂3 (we use here the relation sin Φ = s sin δ Φ). It can be observed that Eqs.
(20) and (21) are Hamilton equations for the canonical variables δ B1 and δ Φ governed by
the Hamiltonian
Q̂
2
(δ B1 ) +Ve f f (δ Φ, ξ ) ,
H(δ B1 , δ Φ, ξ ) = s
(22)
2
where Ve f f = D cos δ Φ − s |α | δ Φ/Q̂ and ξ plays the role of ”time”. Note that Ve f f is a tilted
”washboard” potential with slow ”time”-dependent parameters D and Q̂. Thus, our quasi-steady
states are stable, as long as the effective potential has well defined minima (since cos δ Φ ≈
1 − (δ Φ)2 /2), i.e.
|α | < Ω2 ,
(23)
where Ω2 ≡ −DQ̂ . Eq. (23) expresses the required stability criterion for phase-locking in our
system. We show the dependence of Ω2 on |α | ξ in Fig. 3 for several values of the initial signal
amplitude B2,0 . Note that the most limiting condition on |α | occurs in the transition between the
initial phase locking and the autoresonant stages, i.e. at |α | ξ ≈ −2. The Figure shows that the
#179667 - $15.00 USD Received 12 Nov 2012; revised 20 Dec 2012; accepted 21 Dec 2012; published 15 Jan 2013
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28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 1629
50
0.1
0.2
2
40
Ω
0.05
20
0
−2.1
Ω2
30
0.1
0.01
AR
(b)
−2
−1.9
|α|ξ
10
0
−6
(a)
−4
−2
0
|α|ξ
2
4
6
Fig. 3. Stability function Ω2 versus |α | ξ for different values of the initial signal amplitude
B2,0 (blue: 0.01, red: 0.05, green: 0.1). The black curve represents the value of the analytical expression Eq. (24). Satisfying the criterion |α | < Ω2 continuously is required for the
stability of phase-locking. The inset (b) is a zoom of (a) at the location of the minimum of
Ω2 near |α | ξ = −2. Note that all the curves are almost overlapping.
phase-locked states are stable for the value of α = 0.01 that was used in Fig. 1, but for the case
α = 1 illustrated in Fig. 2, they become unstable for |α | ξ → −2. The spatial frequency Ω is
manifested by the small spatial modulations around the slowly evolving quasi-steady state that
are seen in the examples in Fig. 1. A simple expression for the spatially-dependent frequency
Ω in the autoresonant stage |α |ξ > −2 can be obtained by substituting Eqs. (16)-(17) into the
definition of Ω2 i.e.,
2|α |ξ
(|α | ξ )2 + 12
(|α | ξ )2
+ 4.
(24)
3
3
The minimal oscillation frequency is at |α | ξ = −2 and after this point there is a monotonic
increase in the spatial frequency, as seen in Fig. 1.
Ω2 ≈
4.
+
Conclusions
In summary, we have derived in this article a theory of three-wave mixing in a nonuniform
medium. We have shown that for a medium with weak nonuniformity, a spatially-continuous
phase-locked evolution is established automatically. This evolution is accompanied by a monotonic conversion of the energy between the interacting waves and consequently, it is possible to
achieve complete pump depletion. Our theory could be applied in various fields of physics and
engineering, e.g. Optical Parametric Amplification (OPA) in nonlinear χ (2) media. An estimation for the required values of the design parameters of an autoresonant OPA is provided in the
Appendix.
This work was supported by the FQRNT (Le Fonds Québécois de la Recherche sur la Nature et les Technologies) and NSERC (Natural Sciences and Engineering Research Council of
Canada). O. Y. and L. C. wish to acknowledge the FQRNT MELS fellowship program (Files
168900 and 168739, respectively). M. C. acknowledges the support of the IOF People Programme (Marie Curie Actions) of the European Union’s FP7-2012, KOHERENT, GA 299522.
#179667 - $15.00 USD Received 12 Nov 2012; revised 20 Dec 2012; accepted 21 Dec 2012; published 15 Jan 2013
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28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 1630
5.
Appendix - optical parametric amplification
In order to simplify the design of autoresonant OPAs, we provide in this appendix explicit
formulas in which we specify the required properties of the χ (2) medium in which the TWM
process takes place as well as the experimental setup conditions. We begin by expressing the
normalized nonuniformity rate α and the position |α | ξ in terms of the original physical parameters. From our definitions after Eq. (2) we find that the normalization parameter for length
is
cε n
1,avg n2,avg n3,0 λ1 λ2
0
·
l=
2
8π
de2f f I3,0
1/2
,
(25)
where we used the relations ω j /k j = c/n j and c/ω j = λ j /2π . Here n j and λ j denote the material refractive index and the vacuum wavelength at frequency j, I3,0 = 2n3,0 cε0 |A3,0 |2 is the
initial pump intensity, ε0 is the vacuum permittivity and c is the speed of light in vacuum [25].
By substituting Eq. (25) into the definitions after Eq. (8) and using some algebraic transformations, we see that
l2 (26)
Δk f in − Δkini ,
α=
L
and
Δk
lΔk L
,
(27)
ξ=
= ·
α
l Δk f in − Δkini
therefore
|α | ξ = l · Δk · sign Δk f in − Δkini .
(28)
Here Δkini and Δk f in are the wavevector mismatch at the initial and the final boundaries of the
χ (2) medium, separated by a distance L.
In the previous sections, we have seen that the most significant part of the autoresonant
dynamics occurs between |α | ξ = −2 (which is required to guarantee initial phase-locking)
and |α | ξ = 2 (where according to Eq. (17) B21,2 = 8/9 i.e. about 90% of the initial pump
energy is depleted and transferred to the signal and idler waves). Hence, designing the variation
of the parameters of the system to include the region −2 < |α | ξ < 2 (i.e. |α | ξini < −2 and
|α | ξ f in > 2) will allow efficient
autoresonant amplification. Therefore, we see from Eq. (28)
that it is required that sign Δk f in = sign (Δkini ) and
2
|Δkini | , Δk f in > .
l
(29)
In order to obtain stable phase-locked dynamics in the system everywhere along the propagation, it is required that |α | < Ω2min where Ω2min is the local minimum value of Ω2 that depends
on the initial signal amplitude B2,0 . We see from Eq. (26) that this criterion can be written as
Δk f in − Δkini Ω2
< min
.
(30)
L
l2
For estimating explicitly the required nonuniformity rate in a typical case of interest, we assume
that the dimensionless initial signal amplitude is B2,0 = 0.01 (equivalent to an initial signal to
pump intensity ratio of I2,0 /I3,0 = (n2,0 /n3,0 )2 (ω2 /ω3 ) (B2,0 /B3,0 )2 ≈ 5 · 10−5 for ω2 /ω3 ≈
1/2, n2,0 /n3,0 ≈ 1, recalling that B3,0 ≡ 1). In this case, we see from Fig. 3 that the minimal
value of Ω2 is Ω2min = 0.01 that could be substituted into Eq. (30) to obtain an explicit criterion
for the nonuniformity rate.
#179667 - $15.00 USD Received 12 Nov 2012; revised 20 Dec 2012; accepted 21 Dec 2012; published 15 Jan 2013
(C) 2013 OSA
28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 1631
Considering the fact that the expression Δk(z) ≡ k1 + k2 − k3 = (n1 ω1 + n2 ω2 + n3 ω3 ) /c
depends on the material dispersion relation n j (λ j , z), which is a function of two independent
variables λ j and z, it is not easy to determine analytically what is the required functional form
of the dispersion relation that satisfy the criteria expressed by Eqs. (29) and (30) (note that if the
propagation is within a physically bounded domain, e.g. a waveguide, the necessary conditions
are expressed in terms of a mismatch between the propagation factors that depends also on
the geometry of the system). However, considering a specific experimental setup for which
n j (λ j , z) is known, testing the fulfilment of these criteria numerically is straightforward.
In practice, the desired nonuniformity of the wavevector mismatch may be created by various methods e.g. by nonuniform doping of the nonlinear system. A detailed analysis of feasible
technologies for inducing the required nonuniformity is beyond the scope of the current paper and is left to future studies. Current investigations are ongoing to establish experimental
conditions where autoresonance may strongly boost the efficiency of optical parametric amplification. Preliminary analysis suggests that the proposed autoresonant scheme may allow
complete pump depletion over more than 100 nm bandwidth by properly tuning the refractive
index in few centimetres long standard nonlinear crystals.
#179667 - $15.00 USD Received 12 Nov 2012; revised 20 Dec 2012; accepted 21 Dec 2012; published 15 Jan 2013
(C) 2013 OSA
28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 1632