Propagation of localized optical waves in media with dispersion, in dispersionless media and in vacuum. Low diffractive regime

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by  Lubomir M. Kovachev

Propagation of localized optical waves in media
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0 with dispersion, in dispersionless media and in
0
2 vacuum. Low diffractive regime

n
a
J Lubomir M. Kovachev

4 Institute of Electronics, Bulgarian Academy of Sciences,
1
Tzarigradcko shossee 72,1784 Sofia, Bulgaria

]
s February 2, 2008
c
i
t
p
o Abstract
.
s Wepresentasystematicstudyonlinearpropagationofultrashortlaser
c
pulses in media with dispersion, dispersionless media and vacuum. The
i
s applied methodofamplitudeenvelopesgivestheopportunitytoestimate
y
the limits of slowly warring amplitude approximation and to describe an
h
amplitudeintegro-differentialequation,governingthepropagationofopti-
p
calpulsesinsinglecycleregime. Thewellknownslowlyvaryingamplitude
[
equationandtheamplitudeequationforvacuumarewrittenindimension-

1 lessform. Threeparametersareobtaineddefiningdifferentlinearregimes
v oftheopticalpulsesevolution. Incontrasttopreviousstudieswedemon-
1 strate that in femtosecond region the nonparaxial terms are not small
8
and can dominate over transverse Laplacian. The normalized amplitude
0
nonparaxialequationsaresolvedusingthemethodofFouriertransforms.
2
FundamentalsolutionswithspectralkernelsdifferentfromFresneloneare
.
1 found. One unexpected new result is the relative stability of light pulses
0 with sphericalandspheroidalspatial form,whenwecompare theirtrans-
8
verse enlargement with the paraxial diffraction of lights beam in air. It
0
is important to emphasize here the case of light disks, i.e. pulses whose
:
v longitudinalsizeissmallwithrespecttothetransverseone,whichinsome
i partialcasesarepracticallydiffractionlessoverdistancesofthousandkilo-
X
meters. A new formula which calculates the diffraction length of optical
r pulses is suggested.
a
1 Introduction
For long time few picosecond or femtosecond (fs) optical pulses with approxi-
mately equaldurationin the x, y andz directions(Light Bullets orLB), andfs
opticalpulses withrelativelylargetransverseandsmalllongitudinalsize (Light
DisksorLD)areusedintheexperiments. TheevolutionofsogeneratedLBand
LD in linear or nonlinear regime is quite different from the propagationof light
1
beams and they have drawn the researchers’ attention with their unexpected
dynamical behavior. For example, self-channeling of femtosecond pulses with
power little above the critical for self-focusing [1] and also below the nonlinear
collapse threshold [2] (linear regime) in air, was observed. This is in contra-
diction with the well known self-focusing and diffraction of an optical beam
in the frame of paraxial optics. Various unidirectional propagation equations
have been suggested to be found stable pulse propagation mainly in nonlinear
regime(seee.g. MoloneyandKolesik[3],CouaironandMysyrowicz[4],Chinat
all. [5], for a review). The basic studies in this field started with the so called
spatio-temporal nonlinear Schr¨odinger equation (NSE) which is one compila-
tionbetween paraxialapproximation,the groupvelocity dispersion(GVD) and
nonlinearity[6,7,8,9]. Theinfluenceofadditionalphysicaleffectswerestudied
by adding different terms to this scalar model as small nonparaxiality [15, 18],
plasma defocussing, multiphoton ionization and vectorial generalizations. It is
nothardtoseethatforpulseswithlowintensity(linearregime)inairandgases
the additionalterms asGVD andothersbecome smallandthe basicmodelcan
bereducedtoparaxialequation. Thisisthereasondiffractionofalowintensity
opticalpulsegovernedby thismodelonseveraldiffractionlengthtobe equalto
diffractionof a laserbeam. Onother hand, the experimentalists have discussed
for a long time that in their measurements the diffraction length of an optical
pulse is not equal to this of a laser beam zbeam = k r2, even when additional
diff 0 ⊥
phase effects of lens and other optical devices can be reduced. Here k denotes
0
laser wave-number and r2 denotes the beam waist. Thus exist one deep differ-
⊥
ence between the existing models in linear regime, predicting paraxialbehavior
in gases, and the real experiments.
Thepurposeofthisworkistoperformasystematicstudyoflinearpropaga-
tion of ultrashort optical pulses in media with dispersion, dispersionless media
and vacuum and to suggest a model which is more close to the experimental
results. In addition, there are several particular problems under consideration
in this paper.
The first one is to obtain (not slowly varying) amplitude envelope equation
inmediawithdispersiongoverningtheevolutionofopticalpulsesinsingle-cycle
regime. This problemis naturalinfemtosecondregionwherethe opticalperiod
of a pulse is of order 2 3 fsec. The earliest model for pulses in single-cycle
−
regime suggestedby Brabec and Krausz [20] is obtained after Taylorexpansion
of wave vector k2(ω) about ω . It is easy to show [21] that this expansion
0
divergein solidsfor single-cyclepulses. The higher orderdispersionterms start
to dominate and the series can not cut off. This is the reason more carefully
and accurately to derive the envelope equation before using Taylor series. In
this way we obtain an integro-differential envelope equation where no Taylor
expansion of the wave vector k2(ω), governing evolution of single cycle pulses
in solids.
The second problem is to investigate more precisely the slowly varying en-
velope equations governing the evolution of optical pulses with high number of
harmonics under the envelope. The slowly varying scalar Nonlinear Envelope
Equation(NEE)isderivedinmanybooksandpapers[11,12,13,14,15,16,17].
2
After the deriving of the NEE, most of the authors use a standard procedure
to neglect the nonparaxial terms as small ones. Only some partial nonparaxial
approximationsinfree space[10, 15,18]andopticalfibers [19]werestudied. In
[22]werewritetheNEEindimensionlessformandestimatetheinfluence ofthe
differentlinearandnonlineartermsontheevolutionofopticalpulses. Wefound
thatbothnonparaxialtermsinNEE,secondderivativeinpropagationdirection
and second derivative in time with 1/v2 coefficient, are not small corrections.
InfsregiontheyareofsameorderastransverseLaplacianorstarttodominate.
These equations with (not small) nonparaxialterms are solved in linear regime
[22] and investigated numerically in nonlinear [23]. In this paper we include
GVD term in the nonparaxial model and study also the envelope equation of
electricalfieldin vacuumanddispersionlessmedia. Itis importantto note that
the Vacuum Linear Amplitude Equation (VLAE) is obtained without any ex-
pansion of the wave vector. That is why it work also for pulses in single-cycle
regime (subfemto and attosecond pulses).
Last but not least the nonparaxialequations for media with dispersion, dis-
persionlessmedia andvacuumaresolvedinlinearregimeandnew fundamental
solutions, including the GVD, are found. The solutions of these equations pre-
dict new diffraction length for optical pulses zpulse = k2r4/z , where z is the
diff 0 ⊥ 0 0
longitudinal spatial size of the pulse (the spatial analog of the time duration
t ; z = vt ; v is group velocity). In case of fs propagation in gases and vac-
0 0 0
uum we demonstrate by these analytical and numerical solutions a significant
decreasingofthediffractionenlargementinrespecttoparaxialbeammodeland
a possibility to reach practically diffraction-free regime.
2 From Maxwell’s equations of a source-free, dis-
persive,
nonlinear Kerr type medium to the amplitude
equation
The propagation of ultra-short laser pulses in isotropic media, can be charac-
terized by the following dependence of the polarization of first P~ and third
lin
P~ order on the electrical field E~:
nl
t
P~ = δ(τ t)+4πχ(1)(τ t) E~ (τ,r)dτ =
lin
− −
−Z∞ (cid:16) (cid:17)
t
ε(τ t)E~ (τ,x,y,z)dτ, (1)
−
−Z∞
3
t t t
P~(3) =3π χ(3)(τ t,τ t,τ t)
nl 1− 2− 3−
−Z∞−Z∞−Z∞
E~(τ ,r) E~∗(τ ,r) E~(τ ,r)dτ dτ dτ , (2)
1 2 3 1 2 3
× ·
(cid:16) (cid:17)
whereχ(1) andεarethelinearelectricsusceptibilityandthedielectricconstant,
χ(3) is the nonlinear susceptibility of third order, and we denote r = (x,y,z).
We use the expression of the nonlinear polarization (2), as we will investigate
only linearly or only circularly polarized light and in addition we neglect the
third harmonics term. The Maxwell’s equations in this case becomes:
1∂B~
E~ = , (3)
∇× −c ∂t
1∂D~
H~ = , (4)
∇× c ∂t
D~ =0, (5)
∇·
B~ = H~ =0, (6)
∇· ∇·
B~ =H~, D~ =P~ +P~ , (7)
lin nl
where E~ and H~ are the electric and magnetic fields strengths, D~ and B~ are
the electric and magnetic inductions. We should point out here that these
equations are valid when the time duration of the optical pulses t is greater
0
thanthe characteristicresponsetime ofthe mediaτ (t >>τ ), andalsowhen
0 0 0
the time duration of the pulses is of the order of time response of the media
(t τ ). Taking the curl of equation (3) and using (4) and (7), we obtain:
0 0
≤
1 ∂2D~
E~ ∆E~ = , (8)
∇ ∇· − −c2 ∂t2
(cid:16) (cid:17)
where ∆ 2 is the Laplace operator. Equation (8) is derived without using
≡ ∇
the third Maxwell’s equation. Using equation (5) and the expression for the
linearand nonlinearpolarizations(1) and(2), we canestimate the secondterm
in equation (8) for arbitrary localized vector function of the electrical field. It
is not difficult to show that for localized functions in nonlinear media with and
without dispersion E~ =0 and we can write equation (8) as follows:
∇· ∼
1 ∂2D~
∆E~ = . (9)
c2 ∂t2
4
We will now replace the electrical field in linear and nonlinear polarization on
the right-hand side of (9) with it’s Fourier integral:
+∞
E~ (r,t)= Eˆ~ (r,ω)exp( iωt)dω, (10)
−
−Z∞
where with Eˆ~ (r,ω) we denote the time Fourier transformof the electrical field.
We thus obtain:
t ∞
∆E~ = 1 ∂2 ε(τ t)Eˆ~ (r,ω)exp( iωτ)dωdτ +
c2∂t2  − − 
−Z∞−Z∞
  (11)
t t t ∞
3π ∂2 χ(3)(τ t,τ t,τ t) Eˆ~ (r,ω) 2Eˆ~ (r,ω)
c2 ∂t2 1− 2− 3−
−Z∞−Z∞−Z∞−Z∞ (cid:12) (cid:12)
(cid:12) (cid:12)
exp( i(ω(τ τ +(cid:12)τ )))dωd(cid:12)τ dτ dτ .
1 2 3 1 2 3
× − −
The causality principle imposes the following conditions on the response func-
tions:
ε(τ t)=0; χ(3)(τ t,τ t,τ t)=0,
1 2 3
− − − −
τ t>0; τ t>0; i=1,2,3. (12)
i
− −
That is why we can extend the upper integralboundary to infinity and use the
standard Fourier transform [13]:
t +∞
ε(τ t)exp( iωτ)dτ = ε(τ t)exp( iωτ)dτ, (13)
− − − −
−Z∞ −Z∞
t t t
χ(3)(τ t,τ t,τ t)dτ dτ dτ =
1 2 3 1 2 3
− − −
−Z∞−Z∞−Z∞
+∞+∞+∞
χ(3)(τ t,τ t,τ t)dτ dτ dτ . (14)
1 2 3 1 2 3
− − −
−Z∞−Z∞−Z∞
Thespectralrepresentationofthelinearopticalsusceptibilityεˆ(ω)isconnected
0
to the non-stationary optical response function by the following Fourier trans-
form:
5
+∞
εˆ(ω)exp( iωt)= ε(τ t)exp( iωτ)dτ. (15)
− − −
−Z∞
The expression for the spectral representation of the non-stationary nonlinear
optical susceptibility χˆ(3) is similar :
+∞+∞+∞
χˆ(3)(ω)exp( iωt)= χ(3)(τ t,τ t,τ t)
1 2 3
− − − −
−Z∞−Z∞−Z∞
exp( i(ω(τ τ +τ )))dτ dτ dτ . (16)
1 2 3 1 2 3
× − −
Thus, after brief calculations, equation (11) can be represented as
∞
∆E~ = ω2εˆ(ω)Eˆ~ (r,ω)exp( iωt)dω
− c2 −
−Z∞
∞
+ ω2χˆ(3)(ω) Eˆ~ (r,ω) 2Eˆ~ (r,ω)exp( i(ωt))dω. (17)
c2 −
−Z∞ (cid:12) (cid:12)
(cid:12) (cid:12)
We now define the square o(cid:12)f the lin(cid:12)ear k2 and the generalized nonlinear kˆ2
nl
wave vectors, as well as the nonlinear refractive index n with the expressions:
2
ω2εˆ(ω)
k2 = , (18)
c2
3πω2χˆ(3)(ω)
kˆ2 = =k2n , (19)
nl c2 2
where
3πχˆ(3)(ω)
n (ω)= . (20)
2
εˆ(ω)
The connection between the usual dimensionless nonlinear wave vector k2 and
nl
the generalized one (19) is: k2 = kˆ2 Eˆ~ (r,ω) 2. In terms of these quantities,
nl nl
equation (17) can be expressed by: (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
∞
∆E~ = k2(ω)Eˆ~ (r,ω)exp( iωt)dω
− −
−Z∞
∞
k2(ω)n (ω) Eˆ~ (r,ω) 2Eˆ~ (r,ω)exp( i(ωt))dω. (21)
2
− −
−Z∞ (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
6
Let us introduce here the amplitude function A~(r,t) for the electrical field
E~(r,t):
E~ (x,y,z,t)=A~(x,y,z,t)exp(i(k z ω t)), (22)
0 0
−
where ω and k are the carrier frequency and the carrier wave number of the
0 0
wavepacket. The writing of the amplitude function in this formmeans thatwe
considerpropagationonlyin+z -directionandneglectthe oppositeone. Letus
write here also the Fourier transform of the amplitude function Aˆ~(r,ω ω ):
0
−
+∞
A~(r,t)= Aˆ~(r,ω ω )exp( i(ω ω )t)dω, (23)
0 0
− − −
−Z∞
and the following relation between the Fourier transform of the electrical field
and the Fourier transform of the amplitude function:
Eˆ~ (r,ω)exp( iωt)=
−
exp( i(k z ω t))Aˆ~(r,ω ω )exp(i(ω ω )t), (24)
0 0 0 0
− − − −
Since we investigate optical pulses, we assume that the amplitude function and
its Fourierexpressionaretime -andfrequency-localized. Substituting (33),(23)
and (24) into equation (21) we finally obtain the following nonlinear integro-
differential amplitude equation:
∂A~(r,t)
∆A~(r,t)+2ik k2A~(r,t)=
0 ∂z − 0
(25)
∞
k2(ω) 1+n (ω) Aˆ~(r,ω ω ) 2 Aˆ~(r,ω ω )exp( i(ω ω )t)dω
2 0 0 0
− − − − −
−Z∞ (cid:18) (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
Equation(25)wasderivedwithonlyonerestriction,namely,thattheamplitude
function and its Fourier expression are localized functions. That is why, if
we know the analytical expression of k2(ω) and n (ω), the Fourier integral
2
on the right- hand side of (25) is a finite integral away from resonances. In
this way we can also investigate optical pulses with time duration t of the
0
order of the optical period T =2π/ω . Generally, using the nonlinear integro-
0 0
differential amplitude equation (25) we can also investigate wave packets with
time duration of the order of the optical period, as well as wave packets with a
large number of harmonics under the pulse. The nonlinear integro-differential
amplitude equation (25) can be written as a nonlinear differential equation for
7
the Fourier transform of the amplitude function Aˆ~, after we apply the time
Fourier transformation (23) to the left-hand side of (25) :
∆Aˆ~(r,ω ω )+2ik ∂Aˆ~(r,ω−ω0)
0 0
− ∂z
+ 1+n (ω) Aˆ~(r,ω ω ) 2 k2(ω) k2(ω ) Aˆ~(r,ω ω )=0. (26)
2 − 0 − 0 0 − 0
(cid:18)(cid:18) (cid:12) (cid:12) (cid:19) (cid:19)
(cid:12) (cid:12)
We should note he(cid:12)re the well-kn(cid:12)own fact that the Fourier component of the
amplitude function in equation (26) depends on the spectral difference ω =
△
ω ω , rather than on the frequency, as is the case for the electrical field.
0
−
3 From amplitude equation to the slowly vary-
ing envelope approximation (SVEA)
Equation (25) is obtained without imposing any restrictions on the square of
the linear k2(ω) and generalized nonlinear kˆ2 = k2(ω)n (ω) wave vectors. To
nl 2
obtainSVEA,wewillrestrictourinvestigationtothecaseswhenitispossibleto
approximatek2andkˆ2 asapowerserieswithrespecttothefrequencydifference
nl
ω ω as:
0
−
ω2εˆ (ω) ∂ k2(ω )
k2(ω)= 0 =k2(ω )+ 0 (ω ω )
c2 0 ∂ω − 0
(cid:0) 0 (cid:1)
+1∂2 k2(ω0) (ω ω )2+..., (27)
2 ∂ω2 − 0
(cid:0) 0 (cid:1)
kˆ2 (ω)= ω2χˆ(3)(ω) =kˆ2 (ω )+ ∂ kˆn2l(ω0) (ω ω )+... (28)
nl c2 nl 0 (cid:16) ∂ω (cid:17) − 0
0
To obtain SVEA in second approximation to the linear dispersion and in first
approximation to the nonlinear dispersion, we must cut off these series to the
second derivative term for the linear wave vector and to the first derivative
term for the nonlinear wave vector. This is possible only if the series (27) and
(28) are strongly convergent. Then, the main value in the Fourier integrals
in equation (25) yields the first and second derivative terms in (27), and the
zero and first derivative terms in (28). The first term in (27) cancels the last
term on the left-hand side of equation (25). The convergence of the series
(27) and (28) for spectrally limited pulses propagating in the transparent UV
and optical regions of solids materials, liquids and gases, depends mainly on
the number of harmonics under the pulses [21]. For wave packets with more
than 10 harmonics under the envelope, the series (27) is strongly convergent,
and the third derivative term (third order of dispersion) is smaller than the
8
second derivative term (second order of dispersion) by three to four orders of
magnitude for all materials. In this case we can cut the series to the second
derivative term in (27), as the next terms in the series contribute very little to
the Fourier integral in equation (25). When there are 2 6 harmonics under
−
the pulse, the series (27) is weakly convergent for solids and continue to be
strongly convergent for gases. Then for solids we must take into account the
dispersion terms of higher orders as small parameters. In the case of wave
packets with only one or two harmonics under the envelope, propagating in
solids, the series (27) is divergent. This is the reason why the SVEA does
notgovernthe dynamics of wavepacketswith time durationofthe orderof the
opticalperiodinsolids. Substitutingtheseries(27)and(28)in(25)andbearing
in mind the expressions for the time-derivative of the amplitude function, the
SVEA of secondorder with respect to the linear dispersionand first order with
respect to the nonlinear dispersion is expressed in the following form:
∂A~ ∂A~
∆A~+2ik +2ik k′ =
0 0
∂z ∂t
2
∂2A~ 2 ∂ A~ A~
k k”+k′2 kˆ2 A~ A~ 2ikˆ kˆ′ , (29)
0 ∂t2 − 0nl − 0nl 0nl (cid:12)∂(cid:12)t
(cid:12) (cid:12)
where k0 =(cid:0) k(ω0) and(cid:1) kˆ02nl = kˆn2l(cid:12)(cid:12)(cid:12)(ω0(cid:12)(cid:12)(cid:12)). We will now d(cid:12)efi(cid:12)ne other important
constantsconnectedwiththewavepacketscarrierfrequency: linearwavevector
k k(ω ) = ω ε(ω )/c; linear refractive index n(ω ) = ε(ω ); nonlinear
0 0 0 0 0 0
≡
refractive index n (ω )=3πχ(3)(ω )/ε(ω ); group velocity:
p2 0 0 0 p
1 c
v(ω )= = , (30)
0 k′ ε(ω )+ ω0 1 ∂ε
0 2 ε∂ω
p q
nonlinear addition to the group velocity (kˆ2 )′:
0nl
′ 2k n ∂n
kˆ2 = 0 2 +k2 2, (31)
0nl v 0 ∂ω
(cid:16) (cid:17)
anddispersionof the groupvelocityk”(ω )=∂2k/∂ω2 .All these quantities
0 ω=ω0
allowadirectphysicalinterpretationandwewillthereforerewriteequation(29)
in a form consistent with these constants:
2
∂ A~ A~
∂A~ ∂A~ k v∂n
i +v + n2+ 0 2 (cid:18)(cid:12) (cid:12) (cid:19)=
− ∂t ∂z 2 ∂ω (cid:12)∂(cid:12)t
(cid:18) (cid:19) (cid:12) (cid:12)
 
 
 v v 1 ∂2A~ k vn 2
∆A~ k”+ + 0 2 A~ A~. (32)
2k − 2 k v2 ∂t2 2
0 (cid:18) 0 (cid:19) (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
9
This equation can be considered to be SVEA of second approximation with
respect to the linear dispersion and of first approximation to the nonlinear
dispersion (nonlinear addition to the group velocity). It includes the effects
of translation in z direction with group velocity v, self-steepening, diffraction,
dispersion of second order and self-action terms. The equations (32) and (29),
withandwithouttheself-steepeningterm,arederivedinmanybooksandpapers
[11, 12, 14, 15, 16]. From equations (32), (29), after neglecting some of the
differential terms and using a special ”moving in time” coordinate system, it is
nothardtoobtainthewellknownspatio-temporalmodel. Ourintentioninthis
paper is another: before canceling some of the differential terms in SVEA (32),
we must write (32) in dimensionless form. Then we can estimate and neglect
the smallterms, depending onthe media parameters,the carrierfrequency and
wavevector,andalsoonthe differentinitial shapeofthe pulses. This approach
we will apply in Section 5. As a result we will obtain equations quite different
from the spatio-temporal ones in the femtosecond region.
4 Propagation of optical pulses in vacuum and
dispersionless media
Thetheoryoflightenvelopesisnotrestrictedonlytothecasesofnon-stationary
optical(andmagnetic)response. Eveninvacuum,whereε=1 andP~ =0,we
nl
can write an amplitude equation by applying solutions of the kind (33) to the
wave equation (8). We denote here by V~(x,y,z,t) the amplitude function for
the electrical field E~(r,t) in vacuum:
E~ (x,y,z,t)=V~ (x,y,z,t)exp(i(k z ω t)), (33)
0 0
−
whereω andk againarethe carrierfrequencyandthe carrierwavenumberof
0 0
thewavepacket. Wethusobtainthefollowinglinearequationfortheamplitude
envelope of the electrical field:
∂V~ ∂V~ c 1 ∂2V~
i +c = ∆V~ . (34)
− ∂t ∂z ! 2k0 − 2k0c ∂t2
The vacuum linear amplitude equation (VLAE) (34) is obtained directly from
the wave equation without any restrictions. This is in contrast to the case of
dispersivemedium,whereweusetheseriesofthesquareofthe wavevectorand
we require the series (27) to be strongly convergent. That is why equation(34)
describes both amplitudes with many harmonics under the pulse, and ampli-
tudes with only one or a few harmonics under the envelope. It is obvious that
the envelope V~ in equation (34) will propagate with the speed of light c in vac-
uum. Equation (34) is valid also for transparent media with stationary optical
response ε=const. In this case, the propagating constant will be v =c/√εµ.
10