190
OPTICS LETTERS / Vol. 40, No. 2 / January 15, 2015
Multi-megagauss magnetic field generation
by amplitude modulated surface
plasma wave over a rippled metal surface
Pawan Kumar
Department of Physics, Raj Kumar Goel Institute of Technology, Ghaziabad, UP-201003, India
([email protected])
Received October 7, 2014; revised November 28, 2014; accepted November 30, 2014;
posted December 1, 2014 (Doc. ID 224482); published January 9, 2015
The mechanism of a self-generated megagauss magnetic field by an amplitude modulated surface plasma wave
(SPW) over a rippled metallic surface is proposed. The amplitude modulated SPW exerts a ponderomotive force
on free electrons in the metallic ripple, giving them an oscillatory velocity at the modulation frequency. The osNL
cillatory velocity couples with the electron density in metallic ripple and drives a nonlinear current J⃗
with
NL
∇ × J⃗
≠ 0. This irrotational nonlinear current generates the megagauss magnetic field. © 2015 Optical Society
of America
OCIS codes: (190.4410) Nonlinear optics, parametric processes; (240.4350) Nonlinear optics at surfaces; (240.6680)
Surface plasmons; (240.6690) Surface waves.
http://dx.doi.org/10.1364/OL.40.000190
The surface plasma wave (SPW) plays an important role
in laser coupling to metals [1–5] and overdense plasmas
[6–8]. When the high intensity femtosecond laser gets
mode converted into an SPW, the electric field is greatly
enhanced, leading to a variety of nonlinear effects, e.g.,
multiphoton emission of electrons, ablation of materials,
and harmonic generation [9–14].
A noteworthy feature of high power laser–matter interaction has been the generation of a multigauss magnetic
field because of a variety of mechanisms, e.g., crossed
density and temperature gradients, and a ponderomotive
force across density gradient Weibel instability [15–18].
A large amplitude SPW is also shown to generate quasistatic current in the skin layer of the plasma that produces a magnetic field. As the electron density of the
plasma is increased, the skin depth of the SPW decreases
and so does the generated magnetic field. At a laser intensity of <1016 W∕cm2 , investigated magnetic fields of
the order of 1 MG have been predicted [19]. Two dimensional particle-in-cell simulations for laser–overdense
plasma coupling via resonant excitation of SPW and
quasi-steady state magnetic field generation reveal enhanced laser absorption up to 75%, and the peak value of
the steady magnetic field is 580 MG at an intensity of
∼1019 W cm−2 [8]. An SPW over a rippled metallic surface
is shown to produce in the ripple region a nonlinear
current that has nonzero curl and gives rise to magnetic
field generation [20]. However, in these studies do not
allow temporal variation of laser intensity or consider
a nonperiodic variation.
In this Letter, I develop a mechanism of multi megagauss magnetic field generation by an amplitude modulated SPW over a rippled metallic surface. The physics of
the process is as follows: the amplitude modulated SPW
imparts the oscillatory velocity to electrons to exert a
ponderomotive force on them at modulation frequency.
The ponderomotive force to imparts oscillatory velocity
to electrons which couples with the electron density
rippled and to create a nonlinear current which generates
the magnetic field. This magnetic field is a function of
0146-9592/15/020190-03$15.00/0
laser intensity: as intensity increases the laser intensity
magnetic field increases but our approach is none-relativistic. In other studies, the magnetic field is generated via
resonantly excited nonamplitude modulated SPWs.
Consider a rippled metallic-free space interface
(Fig. 1), x Δ cos qz (where Δ is the height of the ripple) with x < Δ cos qz, representing the metal characterized by relative lattice permittivity εL and free electron
density n0 . As one moves along ẑ in the ripple region,
the electron density shows a periodic variation with z,
hence, following Liu and Tripathi [4], we model the surface ripple as a density ripple with free electron density:
n n0 nq ;
nq n0 iqz
e :
2
(1)
We allow a large amplitude SPW to propagate along ẑ
(cf. Fig. 1). For small Δ, as correspond to the SPW wavelength, the field of the amplitude modulated SPW can be
written as [21]
⃗E 1 A ẑ kz x̂ exp−iωt − kz z expαI x; for x < 0
iαI
kz x̂
exp−iωt − kz z exp−αII x; for x < 0;
A ẑ −
iαII
(2)
where A A0 1 μ cos Ωt − z∕vg α2I k2z − ω2 ∕c2
εL − ω2p ∕ω2 , α2II k2z − ω2 ∕c2 , k2z ω2 ∕c2 εm ∕1 εm ,
εm εL − ω2p ∕ω2 , ωp n0 e2 ∕mε0 1∕2 is the plasma
Fig. 1. Surface plasma wave on metallic rippled surface.
© 2015 Optical Society of America
January 15, 2015 / Vol. 40, No. 2 / OPTICS LETTERS
frequency, e and m are electron charge and mass, εL is
lattice dielectric constant of the metal, and ε0 is the free
space permittivity. We allow A to have slow t, z dependence, A A0 1 μ cos Ωt − z∕vg , where vg is the
group velocity of the SPW, μ is the index of modulation,
and Ω is the modulation frequency. The SPW imparts fast
oscillatory velocity to electrons inside the metal:
⃗vω eE⃗
:
miω
(3)
It also exerts a ponderomotive force on them at the
modulation frequency Ω:
2
k2z
⃗F p − m ∇ ⃗vω · ⃗vω − e
1 2 2αI e2αI x 2A0 1 μ
2
4mω2
αI
× cos Ωt − z∕vg ;
(4)
where we have assumed μ ≪ 1. The ponderomotive force
imparts an oscillatory velocity to electrons:
v⃗ Ω −
F⃗ p
e2 A20 μ1 k2z ∕α2I x̂
αI e2αI x e−iΩt−z∕vg : (5)
miΩ
m2 iΩω2
Had we chosen A A0 cos Ωt − z∕vg , the ponderomotive force would have a 2Ω component and the oscillatory velocity would be
⃗v2Ω −
F⃗ p
e2 A20 1 k2z ∕α2I 2αI x −2iΩt−z∕vg x̂
αI e e
: (6)
2miΩ
4m2 iΩω2
The electron drift beats with the density ripple to produce nonlinear current:
1
NL
J⃗ Ω − nq e ⃗vΩ
2
or
BΩy where εeff εL − ω2p ∕Ω2 , taking ∇× of Eq. (8) and using
Eq. (9), one gets
ε ∂2 B⃗
⃗
μ0 ∇ × J;
∇ × ∇ × B⃗ −∇2 B⃗ − eff
c2 ∂t2
ε
NL
∇2 B⃗ Ω eff
Ω2 B⃗ Ω μ0 ∇ × J⃗ Ω ; x < 0
c2
ω2
∇2 B⃗ Ω 2 B⃗ Ω 0: x > 0.
c
(7)
The current density produces a magnetic field in compliance with Maxwell’s equations:
∇ × E⃗ −
∂B⃗
;
∂t
1 ∂E⃗
NL
;
εeff 2
∇× B⃗ μ0 J⃗
c ∂t
(8)
(9)
(10)
We consider the following two cases:
Case I: Unrippled surface
Using ∂∕∂z Ω∕vg , εeff εL − ω2p ∕Ω2 , Eq. (10) may be
written as
2
∂2 B⃗ Ω
Ω
Ω2 ⃗
NL
2 εeff − 2 BΩ μ0 ∇ × J⃗ Ω
∂x2
c
vg
2
∂2 B⃗ Ω
Ω Ω2 ⃗
x<0
2 − 2 BΩ 0: x > 0.
∂x2
c
vg
(11)
NL
NL
∂ NL
⃗
Since J⃗ Ω ‖x̂, ∇ × J⃗ Ω ŷ ∂z
J Ωx i vΩg J NL
Ωx , BΩ has a ycomponent. Further, the above term varies as e2αI x in
medium I, ∂∕∂x 2αI
μ0 iΩ∕vg J NL
μ0 iΩ∕vg J NL
Ωx
Ωx
BΩy Ω2
≈
:
2
2
Ω
2
4α
ε
−
4α
2
2
I
eff
I
c
v
(12)
g
In a vacuum, the magnetic field falls off as e−βx ,
where β Ω∕vg 1 − v2g ∕c2 1∕2 .
Case II: Rippled surface
NL
J⃗ Ω has z-dependence and varies as of e−iΩt−qΩ∕vg z ,
one may write the magnetic field as
μ0 iΩ∕vg qJ NL
μ iΩ∕vg qenq e2 A20 μ1 k2z ∕α2I αI exp2αI x exp−iΩt − Ω∕vg qz
Ωx 2 0
2 i
h
:
2
2
4α2I Ωc2 εeff − vΩg q
4α2I Ωc2 εeff − vΩg q 2m2 iΩω2
1
NL
J⃗ 2Ω − nq e ⃗v2Ω :
2
191
(13)
The magnetic field has been calculated for moderate
intensity (I L 1016 W∕cm2 ) and the following parameters: jvω j∕c 0.1, nq ∕n0 1∕2, Ω 6 × 1012 rad∕s,
e∕m 1.6 × 1011 , qc∕ωp 1∕20, and εL 9 (gold). The
magnetic field jBΩy j is a function of ω∕ωp and plotted
in Fig. 2. The maximum magnetic field at this intensity
is 3 MG. The figure shows that the magnetic field increases with plasma frequency, but at a higher frequency
it decreases. This is because, as the electron density increases, the skin depth of the SPW decreases and the
generated magnetic field also decreases at a higher
192
OPTICS LETTERS / Vol. 40, No. 2 / January 15, 2015
associated with the generation of the magnetic field
and relevant for many applications of interests.
Fig. 2. Magnetic field if function of frequency.
frequency the coupling of the SPW because of the decreasing of the skin. The rippled surface enhanced magnetic field generation via excitation of the amplitude
modulated SPW.
In conclusion, an amplitude modulated laser excites an
amplitude modulated SPW over a rippled gold surface,
leading to the generation of a magnetic field at the modulated frequency because of the quasi-static nonlinear surface current associated with it. A metallic rippled surface
of suitable wave number gives rise to significant enhancement in the magnetic field. The magnetic field
increases with frequency and decreases at higher frequency. When the magnetic field acquires large value,
the electron cyclotron frequency becomes significant
and affects the propagation of the SPW and nonlinear
phenomena associated with it. A Gaussian laser pulse
of half-period τL could also produce a pulsed magnetic
field. When increasing the intensity of the laser, the magnetic field increased and becomes relevant in many
situations of interests. The SPW can also appear on
flat overdense plasmas by the decay process and is
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