PHYSICAL REVIEW A 82, 013812 (2010)

Hamiltonian structure of propagation equations for ultrashort optical pulses

Sh. Amiranashvili and A. Demircan

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany
(Received 20 December 2009; revised manuscript received 5 May 2010; published 13 July 2010)
A Hamiltonian framework is developed for a sequence of ultrashort optical pulses propagating in a nonlinear
dispersive medium. To this end a second-order nonlinear wave equation for the electric field is transformed into
a first-order propagation equation for a suitably defined complex electric field. The Hamiltonian formulation
is then introduced in terms of normal variables, i.e., classical complex fields referring to the quantum creation
and annihilation operators. The derived z-propagated Hamiltonian accounts for forward and backward waves,
arbitrary medium dispersion, and four-wave mixing processes. As a simple application we obtain integrals of
motion for the pulse propagation. The integrals reflect time-averaged fluxes of energy, momentum, and photons
transferred by the pulse. Furthermore, pulses in the form of stationary nonlinear waves are considered. They yield
extremal values of the momentum flux for a given energy flux. Simplified propagation equations are obtained
by reduction of the Hamiltonian. In particular, the complex electric field reduces to an analytic signal for the
unidirectional propagation. Solutions of the full bidirectional model are numerically compared to the predictions
of the simplified equation for the analytic signal and to the so-called forward Maxwell equation. The numerics
is effectively tested by examining the conservation laws.

DOI: 10.1103/PhysRevA.82.013812 PACS number(s): 42.65.−k, 42.81.Dp, 03.50.De, 45.20.Jj

I. INTRODUCTION The second approach to the ultrashort optical pulses is to

abandon the envelope concept and to operate directly with
The evolution of a wave packet is accurately described
the pulse fields. The simplified model equations are derived
in terms of a complex envelope [1]. The latter results from a
assuming an unidirectional character of pulse propagation
time-scale separation (e.g., when the pulse contains many field instead of SVEA. A recent review is given in Ref. [41]. In
cycles). A slowly varying envelope approximation (SVEA) addition, we mention a short-pulse equation in which the
reduces then the second-order wave equation for the pulse dispersion function is expanded with respect to the inverse
electric field to a more simple first-order nonlinear Schrödinger frequency [42,43] and a more general approach with the
equation (NSE) for the pulse envelope [2–4]. In the frequency Laurent series [44–46]. Another important class of equations
domain, the SVEA assumes that the pulse spectrum is narrow, is given by the (mixed) modified Korteweg–de Vries and
centered around a carrier frequency. However, situations for sine-Gordon models [47–52].
which the SVEA lacks precision are also quite common. For As a rule, such unidirectional propagation equations operate
instance, we mention self-focusing [5,6], optical shocks [7], in the space-time domain, ignore absorption, and use a
steep pulse edge [8], experiments with ultrashort pulses as simplified medium response function. In return, the deduced
optical event horizons [9], and supercontinuum (SC) gener- models often allow for an exact treatment [53–56] or at least for
ation [10]. An important example is that of a few-cycle or a an explicit solitary solution [41,57–63]. Also, many specific
subcycle optical pulse where the spectrum width is comparable solutions to the generalized NSE can be found [33,64–76].
to the carrier frequency [11–16]. In all such situations the NSE The third approach is to derive the pulse propagation
cannot be applied and either a full modeling of Maxwell equa- model in the spectral domain [77–81]. Here, again using the
tions should be undertaken [17–23] or new effective models for unidirectional approximation, one obtains a set of the first-
propagation of spectrally broad pulses should be introduced. order ordinary differential equations for the field harmonics
These models can be developed in different directions. Eω (z). The deduced models are more simple than the full
First, we mention a generalized NSE in which an arbitrary second-order propagation equation and still allow for arbitrary
dispersion profile is approximated by a higher-order Taylor dispersion and spectrum width.
expansion or, more exactly, by a polynomial fit in the frequency In this paper, pulse propagation equations in the spectral
domain. The dispersion is then accounted for by a differential domain are treated from the Hamiltonian point of view.
dispersion operator in the time domain [2,4]. The nonlinear By neglecting medium absorption and considering a one-
term in the generalized NSE is further extended to capture dimensional (z-propagated) setting, we transform the second-
an arbitrary pulse duration [8,24,25]. Furthermore, incorpo- order propagation equation for the electric field E(z,t) into
ration of Raman scattering [26,27], diffraction [24,28,29], the first-order propagation equation for the complex electric
and third-harmonic generation [30] have been discussed. field E(z,t). Positive- and negative-frequency components
The generalized NSE applies to pulse propagation, optical of E(z,t) correspond to the forward and backward waves,
shocks, and SC generation [10,31–38]. It may reproduce the respectively. A Hamiltonian framework is then introduced for
optical field behavior even beyond the validity of the SVEA. the derived propagation equation. To this end we define normal
However, one should note that the dispersion profile for a very variables, A(z,t) and A∗ (z,t). The latter are classical complex
broad spectrum a priori cannot be captured by a polynomial fields and refer to the quantum creation and annihilation
expansion [39,40]. operators.

1050-2947/2010/82(1)/013812(11) 013812-1 ©2010 The American Physical Society

SH. AMIRANASHVILI AND A. DEMIRCAN PHYSICAL REVIEW A 82, 013812 (2010)

By construction, the Hamiltonian is an integral of motion. To proceed, we write the electric field in the frequency
Further integrals are yielded by continuous symmetries of the domain
Hamiltonian. Because of the z-propagated formulation, the  2π
integrals are given by the time-averaged fluxes of the relevant E(z,t) = Eω (z)e−iωt , ω ∈ Z,
ω
T
physical quantities. They provide an effective tool to follow
the numerical solution (e.g., for the SC generation scenarios). where

We also demonstrate that the stationary nonlinear waves can be dt ∗
Eω (z) = E(z,t)eiωt , Eω = E−ω ,
characterized as constrained extrema of the momentum flux. T
Furthermore, reduction of the bidirectional model to a uni- and Eω=0 = 0 in accord with Eq. (4). The linear susceptibility
directional one and further reduction to an envelope equation χ̂ (1) E is defined by a convolution
can be accomplished by simplifying the Hamiltonian function.
The corresponding hierarchy of the first-order propagation (χ̂ (1) E)ω = χ (1) (ω)Eω .
equations is derived. Exemplary solutions of the bidirectional It yields the dielectric constant and the propagation parameter
model are numerically compared to the predictions of the
reduced equations. (ω) = 1 + χ (1) (ω) =  ∗ (−ω),
ω
k(ω) = (ω) = β(ω) + iα(ω) = −k ∗ (−ω),
c
II. DERIVATION
where β(ω) and α(ω) are odd and even functions, respectively.
A. Basic equations Our main concern is the Hamiltonian framework; therefore, a
A formulation of the problem and notations are described small absorption limit is considered. In particular, we neglect
in this section. We consider a sequence of linearly polarized α(ω), assuming that an essential part of the pulse spectrum
electromagnetic pulses propagating along the z axis in a homo- belongs to a transparency window.
geneous dispersive nonlinear medium such that the diffraction The nonlinear susceptibility operator χ̂ (3) is given by the
effects are negligible. The pulse fields E = (E(z,t),0,0) and expression

B = (0,B(z,t),0) are governed by Maxwell equations (χ̂ (3) EEE)ω = χω(3)1 ω2 ω3 ω Eω1 Eω2 Eω3
ω1 +ω2 +ω3 =ω
1
∂z E = −∂t B, − ∂z B = ∂t (0 E + P ), (1) in which summation is performed over suitable (resonance)
µ0
triads {ω1 ,ω2 ,ω3 }. Whenever possible, we abbreviate the sum
where 0 and µ0 are the permittivity and the permeability in the last equation as
of free space, respectively. The induced medium polarization  (3)
P = (P (z,t),0,0) depends on E(z,t) and is determined by a (χ̂ (3) EEE)ω = χ123ω Eω1 Eω2 Eω3 .
123|ω
sequence of nonlocal susceptibility operators χ̂ (i) such that
The condition ω1 − ω2 + ω3 = ω will be indicated as
P (E) = 0 (χ̂ (1) E + χ̂ (2) EE + χ̂ (3) EEE + · · ·), (2) 12̄3|ω. In this way, summations over quads of frequencies
can also be abbreviated. For instance, we will replace
where χ̂ (1) is a linear operator, χ̂ (2) is a bilinear one, and  (3)
 (3)
ω1 −ω2 +ω3 −ω4 =0 χω1 ω2 ω3 ω4 by 12̄34̄| χ1234 .
so on. The power expansion (2) assumes that pulses are
If the dispersion of χ̂ (3) can be ignored, one is left with the
propagating in a weakly nonlinear limit. In addition, an
cubic Kerr medium in which
inverse symmetry is assumed such that P (−E) = −P (E)
and χ̂ (2) = 0. Equations (1) and (2) are reduced to a scalar (χ̂ (3) EEE)Kerr = χ E 3 , χ = const. (5)
nonlinear wave equation
However, for a spectrally broad pulse such an approximation
1 may be invalid and a more general model should be used. For
∂z2 E − 2 ∂t2 (E + χ̂ (1) E + χ̂ (3) EEE) = 0 (3) instance, considering a classical nonlinear oscillator model for
c
electrons, one obtains (Miller’s rule, see Ref. [4])
in which only linear and cubic polarization terms are taken (3)
into account. χ123ω = const × χ (1) (ω1 )χ (1) (ω2 )χ (1) (ω3 )χ (1) (ω).
Integrating Eqs. (1) over time, one sees that the averaged In the following we deal with a general nonlinear susceptibility
electric and magnetic fields are constant along the z axis. For (3)
χ123ω only assuming that it is symmetric with respect to all
simplicity, we assume that the time-averaged fields vanish such permutations of frequencies as suggested by Miller’s rule. The
that (3)
nonlinear absorption is ignored, i.e., χ123ω is a real and even



function of frequencies. The Kerr model (5) is used as an
E(z,t)dt = 0 and B(z,t)dt = 0, (4) illustration.
To proceed, we write the nonlinear wave equation (3) in the
where we use the notation frequency domain

 +T /2 ω2  (3)
dt = dt ∂z2 Eω + β 2 (ω)Eω + 2 χ Eω Eω Eω = 0. (6)
−T /2 c 123|ω 123ω 1 2 3

and T is the period of the pulse sequence. Equation (6) is the starting point of our considerations.

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B. Complex field the t-propagated picture. It leads to a complicated multivalued

In this section we transform Eq. (6) into a first-order expression for the canonical momentum. The z-propagated
propagation equation. To this end, we introduce a complex picture is more simple to deal with because Eq. (11) is already
electric field of first order.
 In this section we introduce a Hamiltonian framework by
E(z,t) = Eω (z)e−iωt , writing Eq. (11) in terms of normal variables. To this end, we
ω change from E(z,t) to a new complex field
where E(z,t) ∈ C and therefore in general Eω differs from 
∗  2µ0 ω2
E−ω = (E−ω )∗ = (E ∗ )ω . The notation A(z,t) = Aω (z)e −iωt
, Eω = Aω , (13)
∗ |β(ω)|
Eω + E−ω Eω + (E ∗ )ω ω
Ĕω = = (7)
2 2 define
is used to separate the real part of E(z,t). The complex µ0 |ω1 ω2 ω3 ω4 |χω(3)1 ω2 ω3 ω4
electric field is defined as a single complex counterpart of Tω 1 ω 2 ω 3 ω 4 = √ , (14)
c2 |β(ω1 )β(ω2 )β(ω3 )β(ω4 )|
two directional variables used in Ref. [81]
and transform Eq. (11) into
ωBω (z) i∂z Eω (z)
Eω (z) = Eω (z) + = Eω (z) − , (8) 
|β(ω)| |β(ω)| i∂z Aω + |β|Aω + 2 T123ω Ăω1 Ăω2 Ăω3 = 0, (15)
where the second representation is obtained from the first 123|ω

equation in (1). Equations (7) and (8) imply where similar to Eq. (7)
1
Ĕω (z) = Eω (z), E(z,t) = E(z,t) + c.c., (9) Aω + A∗−ω
2 Ăω = (16)
2
such that E(z,t) is a complexification of E(z,t). splits contributions of the forward and backward waves.
To get a better insight into Eq. (8), we consider, for (3)
Equation (14) indicates that T1234 and χ1234 have the same
a moment, only a linear medium. For a linear forward
symmetries with respect to permutations of indices.
(backward) wave we have
  Due to its simple and symmetric structure, Eq. (15) can be
±iβ(ω)z ω easily transformed into a Hamiltonian form
Eω (z) ∼ e ⇒ Eω = 1 ± Eω . (10)
|ω| δ
i∂z Aω + H=0 (17)
Therefore Eω>0 and Eω<0 are responsible for the forward and δA∗ω
backward waves, respectively. In particular, contributions of
by defining the following Hamiltonian:
these waves are explicitly split in the relation Eω = 12 (Eω +
∗  
E−ω ). H= |β(ω)||Aω |2 + T1234 Ăω1 Ăω2 Ăω3 Ăω4 . (18)
Returning to the nonlinear case and applying an identity ω 1234|

(∂z2 + β 2 )Eω = (|β| + i∂z )(|β| − i∂z )Eω = (|β| + i∂z )|β|Eω , Equation (17) is a complex representation of the canonical
Hamiltonian equations. The classical fields A(z,t) and A∗ (z,t)
we transform Eq. (6) into the following propagation equation:
are complex canonical variables and refer to the creation and
ω2  (3) annihilation operators (see, e.g., Ref. [84]). By using Eq. (16)
i∂z Eω + |β|Eω + χ Ĕω Ĕω Ĕω = 0. (11) and symmetries of T1234 , one can transform Eq. (18) into the
c2 |β| 123|ω 123ω 1 2 3
form
Equation (11) for Eω (z) is of first order. It looks similar to 
H= |β(ω)|Aω A∗ω + H40 + H31 + H22 , (19)
the unidirectional equation for Eω (z) derived in Ref. [77],
ω
ω2  (3) where contributions of all the possible four-wave-mixing
i∂z Eω + βEω + 2 χ Eω Eω Eω = 0, (12)
2c β 123|ω 123ω 1 2 3 (FWM) processes are explicitly distinguished in the last three
terms,
and can be solved using the same numerical approach.
Equation (11) is, however, exact in the sense the unidi- 1 
H40 = T1234 (Aω1 Aω2 Aω3 Aω4 + c.c.),
rectional approximation was not applied. Both forward and 16 1234|
backward waves exactly fulfill the same first-order propagation
1
model (11) as long as the nonlinearity is calculated from the H31 = T1234 (Aω1 Aω2 Aω3 A∗ω4 + c.c.),
total field (9) (see also Refs. [78,79]). 4
1234̄|
3
C. Hamiltonian framework H22 = T1234 Aω1 A∗ω2 Aω3 A∗ω4 .
8
12̄34̄|
A standard way to obtain first-order Hamiltonian equations
is to perform a Legendre transformation of a second-order One should stress that the system (17) and (18) is an exact
Lagrangian equation [82]. This procedure is discussed in reformulation of Eq. (3) in the weak absorption limit. Such
Ref. [83] for the second-order nonlinear wave equation and a reformulation is useful for some applications which are

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more difficult to address using Eq. (3). Sample applications is an additional integral of motion for the model (17). This
are described in the next section. integral is related to energy transfer. It appears that E is the
period-averaged energy flux. Returning to the real fields in
Eq. (20), one obtains an averaged Poynting vector
III. APPLICATIONS


Hamiltonian formulation of a nonlinear wave equation has EB dt
E= .
many important applications: integrability analysis, conserva- µ0 T
tion laws, stability of solitons, and a spectrum of turbulent Therefore, the simplest vacuum expression for the Poynting
states to name just a few [84–86]. In this section we obtain vector applies also to our nonlinear case. The sum in
conservation laws and relate them to the stationary nonlinear Eq. (20) can easily be evaluated for numerical solutions of the
waves. propagation model (15) and provides a useful tool to control
numerics.
A. Momentum flux
Concerning conservation laws, special care is required C. Stationary nonlinear waves
when the space coordinate serves as an effective time. Con- Stationary nonlinear waves are special solutions of propa-
sider, for instance, a standard continuity equation ∂t ρ + ∂z j = gation equations such that the wave field depends on a single
0 for a physical quantity with the density ρ(z,t) and the variable τ = t − b1 z (retarded time) with a free parameter b1 .
flux density j (z,t) in one space dimension. Normally, the Such solutions propagate with a constant velocity 1/b1 and
conserved integral is given by the “charge” ρ(z,t)dz. For are stationary in the comoving frame. With respect to Eq. (3),
the z-propagated picture we obtain j (z,t)dt = const, i.e., all stationary nonlinear waves can be characterized using the
the time-averaged “current” is constant along the z axis. momentum and energy fluxes.
Returning to the sequence of optical pulses, we conclude
that the period average of the involved variables should not  H and E as functionals acting on a test
Let us consider
function a(τ ) = ω aω e−iωτ such that
depend on the observation point z. Two simple examples are
given by Eqs. (4). By construction, also the Hamiltonian (19)  
H[a] = |β(ω)|aω a∗ω + T1234 ăω1 ăω2 ăω3 ăω4 ,
conserves for Eq. (17). The term ω |β(ω)|Aω A∗ω suggests ω 1234|
that H is related to momentum transfer. It appears that H is
 aω + a∗−ω
the period average of the momentum flux. By transforming E[a] = ωaω a∗ω , ăω = .
Eq. (18) back to real fields in accord with Eqs. (8) and (13), ω
2
one obtains


  We now look for extremal values of H[a] under constrain
0 E + χ̂ (1) E + 12 χ̂ (3) EEE E B 2 dt E[a] = const. To solve this problem, one can set the derivative
H= + .
2 2µ0 T of H[a] − b1 E[a] to zero,
δ δ
For χ̂ (3) = 0, the term in the large parentheses is a known (H[a] − b1 E[a]) = ∗ H[a] − b1 ωaω = 0, (21)
expression for the momentum flux density in a linear δa∗ω δaω
medium [87]. The above expression for H is then the mean where b1 is an unknown Lagrange multiplier.
momentum flux in the nonlinear case. The constrained problem yields both a(τ ) and b1 . After the
solution is found, one can construct
B. Energy flux
A(z,t) = a(t − b1 z) and Aω (z) = aω eib1 ωz .
Further integrals can be obtained from continuos sym-
metries of the Hamiltonian (19). Note, that H is invariant Now, the latter expression for Aω (z) solves Eq. (15) because
under the transformation Aω → Aω eiωs with a free parameter by inserting aω eib1 ωz into the equivalent Eq. (17) one obtains
s. For instance, summation in H40 is performed over quads the extremum condition (21). Therefore, for a given energy
of frequencies such that ω1 + ω2 + ω3 + ω4 = 0. Therefore, flux, a stationary nonlinear wave yields an extremal value of
eiωs factors (appearing in Aω1 Aω2 Aω3 Aω4 ) cancel each other. the momentum flux. Furthermore, unstable nonlinear waves
Similar arguments apply to the other terms in H. correspond to saddle points (see Ref. [88]).
The above continuous transformation is generated by a
differential equation i∂s Aω + ωAω = 0; the latter can be
D. Classical flux of photons
transformed into a Hamiltonian equation
An additional integral of motion appears if contributions
δ  
i∂s Aω + ω |Aω |2 = 0. of both 4 → →
← 0 and 3 ← 1 four-wave processes in the Hamil-
δA∗ω ω
tonian (19) can be neglected (or, strictly speaking, eliminated
using a suitable canonical change of variables as in Ref. [84]).
Following a canonical analog of Noether’s theory (see
In that case, the Hamiltonian reduces to the form
Ref. [82]), we conclude that the quantity
  3
E= ω|Aω |2 (20) H= |β||Aω |2 + T1234 Aω1 A∗ω2 Aω3 A∗ω4 , (22)
ω
8
ω 12̄34̄|

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and the governing Eq. (17) becomes where summations in H31 and H22 are now performed only
over positive frequencies, because Aω<0 = 0 by construction.
3 
i∂z Aω + |β|Aω + T123ω Aω1 A∗ω2 Aω3 = 0. (23) The propagation equation (15) takes the form
4
12̄3|ω
i∂z Aω + β(ω)Aω + S31 + S22 + S13 = 0, (26)
The reduced Hamiltonian (22) is invariant under an ar- where ω > 0 and
bitrary phase shift Aω → Aω eiθ with a free parameter θ .
1 
This continuous transformation is generated by a differential S31 = T123ω Aω1 Aω2 Aω3 ,
equation i∂θ Aω + Aω = 0; the latter can be transformed into 4 123|ω
a Hamiltonian equation 3 
S22 = T123ω Aω1 A∗ω2 Aω3 ,
δ  4
i∂θ Aω + |Aω |2 = 0. 12̄3|ω
δA∗ω ω 3 
S13 = T123ω A∗ω1 Aω2 A∗ω3 .
4
Therefore the quantity 1̄23̄|ω
 At first glance, the unidirectional first-order Eq. (26) is not
N= |Aω |2 (24)
simpler than the bidirectional first-order Eq. (15). However,
ω
Eq. (26) is more convenient for practical uses. To a large
is an integral of motion for the simplified pulse propagation extent one can eliminate the fast dynamics of Aω (z) by
equation (23). By analogy with quantum mechanics, N is transition to a moving frame of reference. Letting V be
proportional to a flux of photons. By transforming Eq. (24) the velocity parameter, we introduce Bω (z) = Aω (z)eiωz/V .
back to real fields in accord with Eqs. (8) and (13), one obtains It is easy to see that Bω (z) is also governed by Eq. (26)
an expression for the period average of the classical photon but with the Doppler-shifted propagation constant β(ω) →
flux β̃(ω) = β(ω) − ω/V . Dynamics in the moving frame is slow
 1  0 (ω)Eω E ∗  if β̃
β. This transformation is especially useful when phase
Bω Bω∗ and group velocities are close to each other for the frequencies
N= ω
+ .
ω
|β(ω)| 2 2µ0 of interest.

Note, that N is infinite when conditions (4) are violated. If it is B. Simplified nonlinear response
not the case, Eq. (24) provides a further useful tool to control
numerics for Eq. (23). A useful and simple propagation equation results if one
can neglect both the backward waves and the 3 → ← 1 four-
IV. PROPAGATION EQUATIONS wave processes. Removing H31 from Eq. (25), one obtains the
following Hamiltonian:
In this section we describe how common unidirectional  3
and envelope propagation equations can be derived from the H= β(ω)Aω A∗ω + T1234 Aω1 A∗ω2 Aω3 A∗ω4 , (27)
Hamiltonian function (19). ω>0
8
12̄34̄|

and the corresponding propagation equation

A. Unidirectional approximation
3 
As cited in the Introduction, propagation equations for i∂z Aω + β(ω)Aω + T123ω Aω1 A∗ω2 Aω3 = 0. (28)
short pulses are usually derived using the unidirectional 4
12̄3|ω
approximation instead of the SVEA. In this section we
Here ω > 0 and both summations are performed over positive
explain how the unidirectional approximation applies with
frequencies. Equations (27) and (28) are unidirectional coun-
respect to the Hamiltonian equation (17). In accord with
terparts of the bidirectional Eqs. (22) and (23), respectively.
Eq. (10), forward and backward waves correspond to positive-
The Hamiltonian (27) is invariant with respect to phase
and negative-frequency components of E(z,t). Neglecting the
shifts and (in addition to H and E) the classical photon flux N
backward wave, we obtain
is conserved. For a numerical solution, Eq. (28) can be trans-
Eω>0 = 2Eω ,  formed to a both moving and oscillating frame of reference. Let
⇒ E(z,t) = 2 Eω (z)e−iωt .
Eω<0 ≈ 0, us assume that for the frequencies of interest the propagation
ω>0 constant β(ω) is closely approximated by a linear function
In other words, the complex E(z,t) becomes an analytic signal β(ω) ≈ β0 + β1 (ω − ω0 ), where ω0 is a reference frequency
corresponding to the real E(z,t). Considering Eq. (19), we see and β0 , β1 are fit parameters. Introducing a new variable
that H40 can be ignored because the condition ω1 + ω2 + ω3 +
Aω (z) = Bω (z)ei[β0 +β1 (ω−ω0 )]z ,
ω4 = 0 cannot be satisfied for positive frequencies. Therefore
the Hamiltonian (19) can be written in the form one derives that Bω (z) is also governed by Eq. (28), but with
 the new propagation constant
H= β(ω)Aω A∗ω + H31 + H22 , (25)
ω>0 β̃(ω) = β(ω) − β0 − β1 (ω − ω0 )
β(ω), (29)

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such that evolution of Bω (z) is slow and more convenient for and obtain
the numerical treatment. 3(ω0 + )χ0
For a spectrally narrow pulse with the carrier frequency ω0 , i∂z ψ + β̃(ω0 + )ψ + (|ψ|2 ψ) = 0,
8cn(ω0 + )
a natural choice is β0 = β(ω0 ) and β1 = β  (ω0 ). This obser-
vation suggests that Eq. (28) is closely related to the envelope where β̃ is defined by Eq. (29).
NSE while being a nonenvelope model and allowing for arbi- The resulting equation, in which n(ω0 + ) is often
trary β(ω). The relationship is investigated in the next section. replaced by n(ω0 ), is the generalized NSE for ultrashort
pulses [8,24,25]. In the reduction of Eq. (30), we only
C. Envelope equation approximate the operator χ (3) . For instance, the NSE and
In this section we demonstrate that the generalized NSE is the field based Eq. (30) are trivially equivalent for the Kerr
just a few approximation steps away from the unidirectional medium. This illustrates why the commonly used generalized
Eq. (28). To this end, we return to the complex electric field NSE reproduces ultrashort pulse propagation observed exper-
(analytic signal for the case at hand) in Eq. (28) by using imentally in photonic crystal fiber, far beyond the validity of
definition (13) the SVEA [10]. Furthermore, one immediately obtains H, E,
and N in terms of the envelope.
3ω 
i∂z Eω + β(ω)Eω + χ123ω Eω1 Eω∗2 Eω3 = 0. (30)
8cn(ω) V. SIMULATIONS OF PULSE PROPAGATION
12̄3|ω

Equation (30) is subject to three conservation laws: Our first objective in this section is to present numerical
solutions of the first-order propagation equation generated by
 0 |Eω |2 30  (3) the full bidirectional Hamiltonian (19). Second, we compare
H= n2 (ω) + χ1234 Eω1 Eω∗2 Eω3 Eω∗4 ,
2 32 the bidirectional model to a simpler propagation equation
ω>0 12̄34̄|
generated by the Hamiltonian (27) and to the so-called
 0 |Eω |2 forward Maxwell equation [77]. To highlight complex and
E= cn(ω) ,
2 comprehensive propagation dynamics, we regard intense
ω>0
ultrashort pulses propagating in a photonic crystal fiber to
 cn(ω) 0 |Eω |2
N= , generate a SC, which is characterized by a dramatic spectral
ω>0
ω 2 broadening. Throughout this section we neglect the dispersion
of the nonlinear suceptibility χ̂ (3) and consider instantaneous
expressing period-averaged fluxes of momentum, energy, and nonlinear polarization (5).
photons, respectively. The bidirectional propagation equation corresponding to
To derive NSE, we introduce an envelope (z,t) for some the Hamiltonian (19) is transformed into the bidirectional
reference frequency ω0 model for the complex field E(z,t) (BMCF)

(z,t) =  (z)e−i t ,  (z) = Eω0 + (z), ω2 χ
i∂z Eω + |β(ω)|Eω + [(E + E ∗ )3 ]ω = 0, (31)
8c2 |β(ω)|
such that where the real optical electric field E(z,t) = Re[E(z,t)].
 BMCF describes both the third-harmonic generation and the
E(z,t) = Eω0 + (z)e−i(ω0 + )t = (z,t)e−iω0 t . self-steepening effect for interacting forward and backward
waves.
The real electric field is expressed as With reasonably given initial conditions, the solutions of the
BMCF facilitate then a direct comparison with results obtained
E(z,t) = Re[E(z,t)] = 12 (z,t)e−iω0 t + c.c., by the forward Maxwell equation (12) (FME, Refs. [77–79]),
which for the case at hand reads
in accord with the definition of the envelope [1]. Note, that
ω2 χ
SVEA is avoided because the transformation from E(z,t) i∂z Eω + β(ω)Eω + (E 3 )ω = 0. (32)
to (z,t) is a trivial change of variables. Returning now 2c2 β(ω)
to Eq. (30), we change to  (z), approximate χω(3)1 ω2 ω3 ω by A further comparison is made with the simplified forward
χ0 = χω(3)0 ω0 ω0 ω0 , and obtain model for the analytic signal (FMAS), corresponding to the
Hamiltonian (27). For Eω>0 (z), the FMAS reads
3(ω0 + )χ0
i∂z  + β(ω0 + ) + (||2 ) = 0. 3ω2 χ
8cn(ω0 + ) i∂z Eω + β(ω)Eω + (|E|2 E)ω>0 = 0. (33)
8c2 β(ω)
Changing to
Here Eω<0 (z) = 0 by construction and E(z,t) is an analytic
 (z) = ψ (z)ei(β0 +β1 )z , signal for E(z,t). The field-based FMAS can be transformed
into the envelope-based NSE for arbitrary pulse widths without
we perform a standard transformation to a moving frame using the SVEA.
The period-averaged momentum flux H and the period-
(z,t) = ψ(z,τ )eiβ0 z , τ = t − β1 z, averaged energy flux E [Eqs. (19) and (20)] are used as control
E(z,t) = 12 ψ(z,t − β1 z)ei(β0 z−ω0 t) + c.c., parameters for the accuracy of the solutions of BMCF. The

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HAMILTONIAN STRUCTURE OF PROPAGATION . . . PHYSICAL REVIEW A 82, 013812 (2010)

classical photon flux N [Eq. (24)] is used as a further control where β2 = β  (ω0 ). The input soliton order N equals
parameter for FMAS. To assure conservation of E, H, and (LD /LNL )1/2 . For our simulations we choose 0 such that
N for an equidistant mesh of time points, we need at least N = 3.54 for the 50-fs pulse and N = 2.24 for the 10 fs pulse.
t = 0.2 fs. Depending on the initial pulse width, we have to
use a resolution of 214 and 215 harmonics for a periodic time B. Backscattered optical field components
window T = 3.5 ps and T = 7 ps, respectively. Several test
To investigate numerically the effect of backscattered com-
calculations were performed for a better resolution, 217 . The
ponents of the optical field in the case of nearly unidirectional
increase of the resolution does not affect the results.
propagation, we regard the BMCF. Forward propagation of the
input pulse is justified by the initial conditions
A. Numerical procedure
[Eω>0 (z) = 2Eω (z)]z=0 and Eω<0 (z)|z=0 = 0,
Here, the direct split-step Fourier approach [3] either
requires very small space steps or lacks precision for a few- where we neglected initial components of backward propagat-
cycle optical pulse and relatively long (e.g., 1 cm) propagation ing waves.
distance, such that the integrals of motion do not conserve. For Figure 1 shows the density plots in the (ω,z) plane of the
our numerics we use, therefore, a de-aliased pseudospectral spectral evolution for positive and negative frequencies for
method. The latter originates from the computational fluid an input pulse with t0 = 50 fs and t0 = 10 fs, respectively.
dynamics [89] and provides a numerical implementation in an The spectra are shown on a logarithmic scale. The spectral
very efficient and accurate manner. The method calculates all broadening of the 50-fs pulse [Fig. 1(a)] in the range between
linear operators and derivatives in the frequency domain and
performs the nonlinear multiplications in the time domain,
with the transformations between the domains achieved by
the fast Fourier transform. The integration for the linear
and nonlinear part is performed in the frequency domain
by a precise Runge-Kutta integration scheme of order eight
with adaptive step-size control, depending on the accuracy as
described in Ref. [90].
The fiber parameters of the highly nonlinear microstruc-
tured fiber are taken from [91]. The propagation constant
β(ω), traditionally approximated by polynomials in ω − ω0 ,
quickly diverges for large frequencies. Instead, we use a proper
rational approximation for the refractive index as in [40], which
gives a correct asymptotic of chromatic dispersion for higher
frequencies and avoids unnecessary numerical stiffness. The
real refractive index is then given by

p0 + p 1 ω + · · · + p 5 ω 5
n(ω) =
1 + q1 ω + · · · + q5 ω5
with the following parameters: p0 = 1.006 54, p1 =
−2.314 31 fs, p2 = 1.959 42 fs2 , p3 = −0.678 111 fs3 , p4 =
0.120 882 fs4 , p5 = −0.009 110 63 fs5 and q1 = −2.299 67 fs,
q2 = 1.947 27 fs2 , q3 = −0.673 382 fs3 , q4 = 0.120 015 fs4 ,
q5 = −0.009 051 04 fs5 . The value of the nonlinear suscepti-
bility χ can be obtained from the nonlinear refractive index n2
by χ = 83 n(ω0 )n2 . For the input pulse electric field we choose

E(z,t)|z=0 = 12 [0 cosh−1 (t/t0 )e−iω0 t + c.c.],

having a central angular frequency ω0 and a hyperbolic-secant

shape for the initial envelope with amplitude 0 and width
t0 . In the following we study the nonlinear propagation of
a 50- and 10-fs pulse, injected at a central frequency ω0 =
2.325 48 PHz, corresponding to a pump wavelength λ0 =
810 nm in the vicinity of the zero dispersion wavelength in
the anomalous dispersion regime. The input pulse amplitude,
width, and the fiber parameters determine the dispersion length
LD and the nonlinear length LNL such that FIG. 1. (Color online) Evolution of χ |Eω |2 in the (ω,z) plane for
the 50-fs pulse (a), and the 10-fs pulse (b). The spectra are shown
|β2 | ω0
L−1
D = and L−1
NL = n2 02 , in logarithmic scale (dB). Negative frequencies reflect backscattered
t02 c optical field components.

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SH. AMIRANASHVILI AND A. DEMIRCAN PHYSICAL REVIEW A 82, 013812 (2010)

χ 1 2E z 5.05 cm 10 6
χ 2
0.01 ω
12
10
0
18
0.01 10
Τ fs Ω PHz
200 100 0 100 200 300 10 5 0 5 10 FIG. 2. Exemplary profiles of the
12 z 2.5 cm 2 forward propagating electric field and
χ χ
6
E 10
ω
0.01
12
the corresponding spectra of the full
10
0 complex field for a 50-fs pulse. The
18
0.01 10 temporal profiles are shown in a for-
Τ fs Ω PHz
ward comoving frame with τ = t −
200 100 0 100 200 300 10 5 0 5 10 β1 z.
χ 12
E z 0.5 mm 10 6
χ 2
0.01 ω
12
10
0
18
0.01 10
τ fs ω PHz
200 100 0 100 200 300 10 5 0 5 10

0–5 PHz exhibits the typical scenario for SC generation by consequently smaller. The soliton order is lower, so that SC
soliton fission, demonstrated by modeling both the FME [77] generation by the known soliton fission process is less efficient.
and the generalized NSE [10]. Snapshots of the temporal Snapshots of the temporal shapes for selected propagation
shapes for selected propagation distances and corresponding distances and corresponding spectra are presented in Fig. 3.
spectra are presented in Fig. 2. For ω > 5 PHz the spectral evo- The strong initial spectral broadening is mainly generated
lution is determined by third-harmonic generation, featuring by self-phase modulation and soliton compression and the
similar broadening for the third-harmonic components. The spectrum reaches an octave coverage after only a few mm. In
properties of backscattered components of the optical field are the time domain we observe one main pulse in the anomalous
represented by negative frequencies. From the beginning of dispersion regime, forming one fundamental soliton, whereas
the propagation the main pulse and the third harmonic lead to the rest of the incident pulse in the normal dispersion regime
the excitation of two extremely weak counterparts of backward disperses. After the fundamental soliton is generated, the
waves on the negative side of the spectrum. The strong initial spectrum becomes almost frozen (z > 20 mm). Again there is
contraction of the main pulse and the fission of fundamental an excitation of two main components of backscattered light
solitons results in an enhanced spectral broadening, reflected with negative frequencies and the spectral broadening on the
also by the negative frequencies, which evolve in an analogous negative side follows the pump pulse compression and the
manner (see also Fig. 2 at z = 2.5 cm). After the breakup of accompanied spectral broadening of the third harmonics. The
the higher-order soliton (Fig. 2 at z = 5.05 cm) the spectral FWM interaction of radiated dispersive waves and ejected
width is already saturated, but FWM generates complicated fundamental solitons is low and the spectral profile of the
substructures, which leads to a complicated substructure for backward waves retains an unruffled shape. However, the
all negative frequencies. negative spectral components generated by the nonlinear
Also for the 10-fs pulse [Fig. 1(b)], the nonlinear coupling coupling remain negligibly small and the propagation of the
causes a similar spectral evolution for negative frequencies forward waves remains unaffected by their interaction.
as for the positive frequencies even though the input energy In Fig. 4 we present the temporal evolution of the backward
is reduced and the energy shift to negative frequencies is moving waves for the 10-fs pulse (left column) and 50-fs pulse

χ 1 2E z 3.5 cm 10 12
χ 2
0.01 ω

0 24
10
0.01
Τ fs 10 36 Ω PHz
50 0 50 100 150 200 10 5 0 5 10 FIG. 3. Exemplary profiles of the
12 z 1.0 cm 2 forward propagating electric field and
χ E 10 12
χ ω
0.01 the corresponding spectra of the full
0 10 24 complex field for a 10-fs pulse. The
0.01 temporal profiles are shown in a for-
Τ fs 36 Ω PHz
10 ward comoving frame with τ = t −
50 0 50 100 150 200 10 5 0 5 10 β1 z.
χ 12
E z 0.5 mm 10 12
χ 2
0.01 ω

0 24
10
0.01
τ fs 10 36 ω PHz
50 0 50 100 150 200 10 5 0 5 10

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HAMILTONIAN STRUCTURE OF PROPAGATION . . . PHYSICAL REVIEW A 82, 013812 (2010)

χ 1 2E z 7.5 mm χ 1 2E z 1.5 cm
7 7
10 10
7 7
10 10
Τ fs Τ fs
250 200 150 100 50 0 50 500 400 300 200 100 0 100 FIG. 4. Exemplary time pro-
12 z 5.0 mm 12 z 1.0 cm files of the backscattered electric
χ E χ E
7 7 field components for a 10-fs pulse
10 10
(left column) and a 50-fs pulse
7 7
10 10 (right column). The profiles are
Τ fs Τ fs shown in a backward comoving
250 200 150 100 50 0 50 500 400 300 200 100 0 100 frame with τ− = t + β1 z.
χ 12
E z 0.5 mm χ 12
E z 0.5 mm
7 7
10 10
7 7
10 10
τ fs τ fs
250 200 150 100 50 0 50 500 400 300 200 100 0 100

(right column) for selected propagation distances. The z values C. Comparison with unidirectional models
are chosen such that the backward waves are not superposed In this section we concentrate on the dynamics of the
with the forward waves in our periodic time window and forward-propagating waves, represented by positive frequen-
well-separated pulses can be observed. The two backward cies. The spectral evolution of the electric field components
propagating field components have their center at t = 0 at the of the 10-fs pulse and the 50-fs pulse obtained by the BMCF
beginning of the propagation with the same temporal profile (solid black line), the FME (solid gray line), and the FMAS
and width as the incident forward propagating pump wave and (dashed black line) are presented in Fig. 5.
the third harmonics, but with amplitudes, which are several The solutions of the BMCF for positive frequencies do not
magnitudes lower. They propagate in the opposite direction differ globally from the solutions of the FME. The FMAS lacks
with the same velocity as the forward propagating parts, so that 3→ ← 1 four-wave processes and no third-harmonic generation
the backscattered field components of the third harmonic move is observed, but in the spectral range 0–5PHz the same spectra
faster than the backscattered field components of the input are received as by the BMCF or the FWE. The differences
pulse moving with the group velocity β1−1 . The propagation between the three models are more visible in Fig. 6, where the
of the backward waves is mainly affected by the linear part spectra for the 50-fs pulse are shown in logarithmic and linear
of BMCF and the temporal profiles stay unchanged over the scale and around the carrier frequency. For the FMAS there is
regarded propagation distances. no energy shift to backscattered optical field components and
The amplitude of the backscattered components depends on to higher harmonic generation and the main pulse contains
the nonlinear term in BMCF and increases with the nonlinear more energy. The SC generation evolves slightly faster in
susceptibility χ . For the given pulse and fiber parameters the that case (dashed line). A similar deviation exists between
shift of energy to negative frequencies is extremely weak and the BMCF and the FME, whereas the energy difference lies
the impact of backscattered light on the propagation dynamics only in shift of energy to the backward waves.
of the forward wave can be neglected. We demonstrate this in In short, predictions of all three models are very sim-
the next section by comparing the predictions of the BMCF ilar around the pulse carrier frequency. Therefore also the
with simulation of the FME and the FMAS.

χ Eω 2 z 2.5 cm χ Eω 2 z 5.05 cm
10 10
10 10
20 20
10 10
10 30 Ω PHz 10 30 Ω PHz
0 2 4 6 8 10 12 0 2 4 6 8 10 12 FIG. 5. Electric field spectra for
2 z 7.5 mm 2 z 2.5 cm selected propagation distances for a
χ Eω χ Eω
10 10
10 10 10-fs pulse (left) and 50-fs pulse
20 20 (right). Calculations were performed
10 10
with the BMCF (solid black line), the
10 30 Ω PHz 10 30 Ω PHz
FME (solid gray line), and the FMAS
0 2 4 6 8 10 12 0 2 4 6 8 10 12 (dashed black line).
χ Eω 2 z 0.5 mm χ Eω 2 z 0.5 mm
10 10
10 10
20 20
10 10
10 30 ω PHz 10 30 ω PHz
0 2 4 6 8 10 12 0 2 4 6 8 10 12

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SH. AMIRANASHVILI AND A. DEMIRCAN PHYSICAL REVIEW A 82, 013812 (2010)

10 8
χ Eω 2 z 5.05 cm χ Eω 2
9
5 10
9
10 FIG. 6. Detailed comparison of
9
3 10
10 10 the spectra predicted by the BMCF
Ω PHz 1 10 9 Ω PHz (solid black line), the FME (solid gray
11
10
1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.8 2.0 2.2 2.4 2.6 2.8 line), and the FMAS (dashed black
10 8
χ Eω 2 z 2.5 cm χ Eω 2 line) around the carrier frequency for
9
5 10
10 9 a 50-fs pulse in logarithmic scale (left)
3 10 9 and linear scale (right).
10
10
ω PHz 1 10 9 ω PHz
11
10
1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.8 2.0 2.2 2.4 2.6 2.8

simplest FMAS reproduces the most essential features of pulse useful tool to control numerical solutions. They can also
propagation. Even for a relatively long propagation distance, be used to characterize solitons. In particular, a stationary
the transfer of energy for third-harmonics generation and for a nonlinear wave is governed by the following property: the
backscattered wave remains small and, most important, there wave yields an extremal value of the momentum flux for
is no noticeable feedback to the main pulse. The quality of all a given energy flux. We also show that both nonenvelope
presented numerical solutions is effectively controlled by the (unidirectional) and envelope propagation equations can be
conservation laws. derived directly from the Hamiltonian representation.
Finally, we illustrate numerically the propagation dynamics
VI. CONCLUSIONS described by the bidirectional model for the complex field by
Let us summarize our results. Propagation of spectrally calculating supercontinuum generation for ultrashort pulses.
broad ultrashort optical pulses is considered. First, we show The solutions reproduce all essential features seen in a number
how the standard second-order propagation equation can be of experiments [10,92] and in simulations with the generalized
transformed into a first-order model for a properly chosen nonlinear Schrödinger equation [10]. In addition, the effect of
complex electric field. The model looks similar to the first- backscattered optical field components can be investigated.
order unidirectional models reported previously; however, it A comparison with the forward Maxwell equation [77–79]
accounts for both forward and backward waves. These waves and with the simplified equation for the analytic signal
are described by positive- and negative-frequency components exemplifies that under certain circumstances also the latter
of the complex electric field. The latter reduces to an analytic model represents a practical and useful tool for the description
signal for the purely forward waves. of pulse propagation in nonlinear media.
Second, we present the bidirectional first-order propagation
equation as a Hamiltonian one. To this end the so-called
ACKNOWLEDGMENTS
normal variables (classical counterparts of the creation and
annihilation operators) are introduced. Then we obtain con- Helpful discussions with U. Bandelow, U. Leonhardt,
servation laws for the z-propagated picture. They are given by M. Lichtner, and A. Mielke are gratefully acknowledged. This
the time-averaged fluxes of momentum, energy, and photons work was supported by the DFG Research Center MATHEON
transferred by the pulse. The conservation laws provide a under Project No. D 14.

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