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J. Opt. Soc. Am. B / Vol. 27, No. 1 / January 2010
Balakin et al.
Terahertz emission from a femtosecond laser focus in a two-color scheme Alexey V. Balakin,1 Alexander V. Borodin,1 Igor A. Kotelnikov,2 and Alexander P. Shkurinov1,* 1
Department of Physics and International Laser Center, M. V. Lomonosov Moscow State University, Leninskie Gory GSP-1, Moscow 119992, Russia 2 Budker Institute of Nuclear Physics, Lavrentyev Avenue 11, Novosibirsk 630090, Russia *Corresponding author:
[email protected] Received September 3, 2009; accepted October 22, 2009; posted October 28, 2009 (Doc. ID 116676); published December 11, 2009 We critically revise the theory of terahertz emission from a plasma filament induced in a gas media by one or two focusd femtosecond laser pulses. We distinguish a radiation pressure force (RPF) from a ponderomotive force (PF), discuss conditions for one of these forces to be the dominating contribution to the terahertz emission, and also show that the angular distribution of the emitted power critically depends on which of the two forces dominates in a particular experiment. We show that the experimentally observed periodic dependence of the emitted terahertz power on the gas pressure reveals the dominating role of the RPF over the PF, whereas the angular diagram of the emission allows us to determine the predominant direction of the force. We also emphasize that the terahertz emission originated by a transient photocurrent exhibits a different dependency from the phase difference between the first and the second harmonics of the optic laser field, which generally enables the experimental detection of the prevailing mechanism of the terahertz emission from the plasma filament. © 2009 Optical Society of America OCIS codes: 260.3090, 190.2620, 350.5400, 300.2570.
1. INTRODUCTION A particular subject of plasma science is to explain various kinds of electromagnetic radiation ranging from terahertz to x-ray frequencies induced by high-power laser pulses interacting with photoinduced plasma [1,2]. It has been observed that a broadband electromagnetic pulse (EMP) of terahertz radiation can be emitted from a plasma filament produced by an ultrashort (less than 100 fs) high-intensity 共1012 – 1015 W / cm2兲 laser pulse propagating in a gas medium [3–5]. Earlier published theories suggest that the basic mechanism behind the terahertz radiation relies on the effect of a ponderomotive force (PF), which separates electrons from ions spatially within the plasma filament [5,6] and, consequently, generates a nonlinear electric current. Recently, it has been noted that the emitted terahertz power increases manifold if a two-color scheme is used where a fundamental optical wave with the frequency is mixed with its second harmonic (SH) satellite at the doubled frequency 2 [7–9]. The nonlinear electric current gives rise to the plasma filament polarization. Phenomenological models, formulated in terms of nonlinear polarization susceptibilities, attribute the terahertz emission to a four-wave mixing (FWM) rectification process [7–12]. The phenomenology describes polarization properties of the terahertz field reasonably well [9,13] and the fact that the terahertz field is subject to the phase difference between the fundamental and the SH waves. Specifically, it predicts that ETHz ⬀ E2 E2 cos [7,11,14,15] or ETHz ⬀ E2 E2 sin [9]. Within the phenomenological approach it is hardly possible to distinguish whether the cos or the sin dependency is correct but the latter has been confirmed by experimental studies [9,16]. Furthermore, it has been 0740-3224/10/010016-11/$15.00
shown that ETHz ⬀ E2 and ETHz ⬀ E2 for sufficiently low fundamental wave energy (less than 100 J) [8,13]. The understanding of plasma formation through optical breakdown has a paramount value for the unveiling of the physical mechanism of the terahertz emission from an optically generated plasma filament. Earlier dynamic theories of the emission assumed that the photoelectrons, pulled away from gas atoms by strong laser fields, are born with zero macroscopic (i.e., average) velocities [5]. This natural contemplation was disputed in [16,17]. It was suggested that the terahertz emission at a microscopic level originates from a transient photocurrent created by the photoelectrons produced in a symmetrybroken laser field composed of fundamental and phaseshifted SH optic waves. In this paper we present a simple theory based on classical assumptions of the terahertz emission, which takes into account the FWM process. It has led us to the conclusion that the manifold increase of the radiated power, observed in a number of experiments, occurs mainly due to transient photocurrent effect and also due to enhanced plasma production in the process of photoionization, since the multiphoton ionization (MPI) at doubled frequency goes on faster than at the fundamental frequency. Additional enhancement occurs due to the effect of the plasma pressure force [18]. For the first time, to our knowledge, we compute the radiation pressure force (RPF) for a laser pulse that contains both the first harmonic and the SH. We then argue that this newly computed force can produce a larger terahertz emission even if it is smaller than the PF. At a phenomenological level we also add to the theory the effect of the photocurrent by allowing a nonzero initial macroscopic velocity of the photoelectrons. Al© 2010 Optical Society of America
Balakin et al.
Vol. 27, No. 1 / January 2010 / J. Opt. Soc. Am. B
though we are not ready at the moment to compute the initial velocity from a strict quantum theory, we present a simple classical consideration of the photoionization process in a two-color laser field, which might hopefully elucidate major properties of the terahertz emission originated from the photocurrent. In particular, we note that the terahertz emission induced by the photocurrent scales as sin while that induced by the RPF or the PF varies in proportion to cos . In our model the electrons and ions are initially created by the MPI within laser string, and a dipole moment is subsequently induced in the plasma filament via the RPF acting on the electrons and resulting from the velocitydependent Lorentz force and from the PF resulting from spatial inhomogeneity of the laser fields. These forces separate the electrons from the heavy ions both longitudinally and transversely regarding the plasma filament axis on the short time scale of the laser pulse. Behind the laser pulse head the electron fluid oscillates at plasma frequency and generates an EMP propagating away from the plasma filament as first described in [18]. Since plasma frequency is proportional to the square root of the local electron density, and the density varies from point to point within the plasma filament, the frequency spectrum of the terahertz emission ranges from a maximal value down to almost zero. The paper is organized as follows. In Section 2 we compute the RPF and then in Section 3 we compute the PF. Slow motion of the plasma electrons under the action of these forces that spatially separates the electrons from the ions is analyzed in Section 4 and the resulting terahertz emission is computed in Section 5. Section 6 adds to our consideration of the effect of the transient photocurrent. Section 7 specifies experimental apparatus used to validate the theory and our conclusions are summarized in Section 8.
assumed to remain unmovable. The RPF indicated below as G formally originates from a velocity-dependent part of the Lorentz force, i.e., from the second term on the right-hand side (RHS) of Eq. (1). Since an oscillatory velocity of an electron in the field of a laser pulse is of the order of v ⬃ eE / m, and B ⬃ E, one formally finds that G ⬃ mv2 / ⑄, where ⑄ = c / is the reduced laser radiation wavelength. The PF indicated below as F formally originates from the second term on the left-hand side of Eq. (1), which is evaluated as mv2 / a, with a being a gradient length of the laser pulse envelope. Since a Ⰷ ⑄, one might deduce that G Ⰷ F in any circumstances, but this would be a rushed conclusion because the main term mv2 / ⑄ in G turns out to be zero due to phase cancellation. One can suggest although that the next term in an expansion of G over v can be as large as F, alternatively, collisional friction can prevent complete phase cancellation. At the first stage we ignore spatial derivatives in Eq. (1) and drop the term m共v · ⵜ兲v; it will be retained over again in Section 3. Furthermore, spatial dimensions of the laser string are assumed in this section to be so enormous that the laser electromagnetic fields E = EL and B = BL can be approximated as spatially uniform over the transverse profile of the plasma filament. In the rest of this section we sequentially consider a case of mutually parallel harmonics, E2 储 E, and that of perpendicular harmonics, E2 ⬜ E. After having completed the calculations, we shall see that the treatment of these two geometries is sufficient for a comprehensive characterization of the RPF. A. Case E2 ¸ E Let a laser pulse be linearly polarized along the x axis while propagating in air along the positive direction of the z axis so that EL = ELxˆ ,
2. RADIATION PRESSURE FORCE To calculate the RPF we extend a model first elaborated in [18], where the force was computed for a single-color laser field. In contrast to the PF to be analyzed in Section 3, the RPF does not explicitly depend on the gradient sizes of the laser beam and, therefore, it dominates for a relatively wide beam. In a single-color scheme, the RPF is directed along the laser pulse path [18], whereas a transverse component of the force emerges when a laser pulse at the fundamental frequency is mixed with that at the doubled frequency 2. Following [3,4,18,19], we employ an electron-fluid model of the plasma in the presence of the electric and magnetic fields E and B, which is governed by the equation m
v
e + m共v · ⵜ兲v = eE + v ⫻ B − ␥mv, t c
共1兲
for the fluid velocity v of the electrons, where e and m are the charge and the mass of the electron, and −␥mv approximates a friction force due to electron scattering by heavy constituent particles (counting both ions and neutral atoms); throughout the paper, the heavy particles are
17
BL = ELyˆ ,
共2兲
where EL = 21 E + 21 E2 + c.c., and E = E1共t − z/vg共兲兲exp关ik共兲z − it兴, E2 = E2共t − z/vg共2兲兲exp关ik共2兲z − i2t兴, are the laser beam electric fields at the basic and doubled frequencies, with t1⬘ = t − z / vg共兲 and t2⬘ = t − z / vg共2兲 being the retarded times in the reference frames moving with the laser pulse group velocities vg共兲 and vg共2兲, respectively. The parameter 1 / c in Eq. (1) formally tracks the order of perturbation theory and reflects the fact that the velocity-dependent Lorentz force is of the order of v / c Ⰶ 1 smaller than the dominant force due to the electric field under nonrelativistic conditions appropriate here. Then, to zeroth and first orders in the expansion parameter 1 / c, Eq. (1) yields the triplet of equations, m
vx t
= eEL − e
m
vy t
vz c
EL − ␥mvx ,
= − ␥mvy ,
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J. Opt. Soc. Am. B / Vol. 27, No. 1 / January 2010
vz
m
=e
t
vx c
Balakin et al.
EL − ␥mvz .
共3兲
Gx =
vx ⬇
冋
E
2m 共␥ − i兲
+
Eⴱ 共␥ + i兲
+
E 2 共␥ − i2兲
E2ⴱ
+
共␥ + i2兲
册
Gz =
冓 冔 e
c
EL
␥兩E兩2
e2 =
␥兩E2兩2
e2
2mc 共␥2 + 2兲
+
2mc 共␥2 + 42兲
.
c
冉 冊冋 2
e
=
2mc
+
+
+
+
+
+
+
E2ⴱ
共␥ − i兲 ␥ E 2
共␥ + i2兲 ␥ Eⴱ
Eⴱ
E
+
E 2
E Eⴱ
共␥ + i兲 共␥ + i2兲 Eⴱ
E2ⴱ
共␥ + i兲 共␥ + i3兲 E2ⴱ
,
共4兲
2
2mc
+
E
E
E 2
2mc
共␥ − i兲 共␥ − i3兲
E 2
E
共␥ − i2兲 共␥ − i3兲
册 冉 冊冋 册 e
+
2mc
2
E 2
关E2 E2ⴱ + c.c.兴,
共5兲
B. Case E2 E Now let the SH be polarized perpendicularly to the first harmonic so that 1
1
EL = 2 Exˆ + 2 E2yˆ + c.c.,
E 2
共␥ − i2兲 共␥ − i4兲
.
The first pair of square brackets here comprises low (zero) frequency terms and the second pair contains the first harmonic, i.e., the terms varying as e+it or e−it, etc. Putting the second and the third pairs into the term e共vz / c兲EL of the equation for vx gives the low frequency force,
1
共6兲
BL = − 2 E2xˆ + 2 Eyˆ + c.c. The equations of motion now take the form m
m
m
vx
e e vz = 共E + Eⴱ 兲 − 共E + Eⴱ 兲 − ␥mvx , t 2 2 c
vy
e e vz = 共E2 + E2ⴱ 兲 − 共E2 + E2ⴱ 兲 − ␥mvy , t 2 2 c
vz t
=
e vx 2 c
共E + Eⴱ 兲 +
e vy 2 c
共E2 + E2ⴱ 兲 − ␥mvz .
共7兲
Using the same method as in Subsection 2.A for solving these equations recovers formula (4) for the longitudinal force Gz. The transverse force, Gy =
冓
=−
共␥ − i兲 共␥ − i2兲
2
e
3␥2
8m2c2 共␥2 + 2兲共␥2 + 42兲
⬇ E
EL
−
e vz 2 c e3
E2ⴱ
共␥ − i2兲 共␥ − i兲
+
共␥ + i2兲 共␥ + i4兲
共␥ − i2兲 ␥
Eⴱ
册 冉 冊冋 册 冉 冊冋
E2ⴱ
E2ⴱ
E 2
=−
共␥ − i兲 共␥ + i兲
e
Eⴱ
+
E
E 2
+
共␥ + i2兲 共␥ − i3兲 E2ⴱ
2
2mc
+
E
共␥ + i兲 ␥
册 冉 冊冋
共␥ + i2兲 共␥ + i兲 Eⴱ
Eⴱ
e
共␥ + i兲 共␥ − i兲 E2ⴱ
+
c
1
representing the RPF along the direction of the laser pulse propagation. Formula (4) has been derived in [18]. It should not be confused with the light pressure force fT = T共兩E兩2 + 兩E2兩2兲 / 8, where T = 共8 / 3兲共e2 / mc2兲2 is the Thomson cross section of an electromagnetic wave scattering by a free electron. The light pressure force can be derived from Eq. (1) with the radiative reaction force 共2e2 / 3c3兲v¨ substituted instead of the friction force −␥mv. In most circumstances, the light pressure force is negligibly small, fT Ⰶ G z. Straightforward integration of Eq. (3) for vz yields five harmonics, from zeroth to fourth, varying as e±i0t, e±it, e±i2t, e±i3t, and e±i4t, respectively, vz
vz
acting in the direction of the laser field polarization.
The equation for vy means that vy = 0, and the electron motion is limited to the xz plane. Next, the inspection of the term e共vx / c兲EL in the equation for vz reveals that it has the low frequency part, vx
−e
e3
Under the assumption that the electric field envelope EL varies slowly on the carrier time scale −1, the equation for vx is approximately solved as e
冓 冔
共E2 + E2ⴱ 兲
冋
冔
E2 E2ⴱ
8m2c2 共␥ − i兲共␥ − i2兲 e3
16m2c22
关E2 E2ⴱ + c.c.兴,
+ c.c.
册 共8兲
turns out to be parallel to E2. As shown in Subsection 2.A, this is also the case for E2 储 E. However, in a general case the transverse force is not exactly parallel to E2 since Gx / Gy ⫽ E2,x / E2,y. For E2,x = E2,y the ratio Gx / Gy ⬇ 3␥2 / 22 is expected to be small. In other words, the effect of the SH is stronger when it is polarized differently from the first harmonic. The longitudinal force Gz is small as compared with Gy provided that the SH is strong enough, E2,y Ⰷ 4␥mc / e. To evaluate ␥ we assume that the electron–atom collisions dominate over the electron–ion collisions, which is the case for sufficiently low ionization of the ambient gas in the plasma filament. Then ␥ = naeave, where ea ⬃ 10−16 cm−3 is the cross section of the electron scattering by neutral atoms, na is the atoms’ density, and the average velocity ve of the electrons is of the order of their oscillatory velocity v ⬃ eE / m. With these assumptions used, the condition E2 Ⰷ 4␥mc / e is equivalent to the inequality E2 / E Ⰷ 4cnaea / ⬃ 0.2na / natm, where natm ⬇ 3 ⫻ 1019 cm−3 is the air density at normal conditions and = 2 ⫻ 1015 s−1.
Balakin et al.
Vol. 27, No. 1 / January 2010 / J. Opt. Soc. Am. B
The condition can be met in a rarefied gas. Furthermore, if it is not satisfied, the transverse force can still produce a more powerful radiation from the plasma filament as shown in Section 5. Following [20,21], many papers refer the estimation ␥ ⬇ 1012 – 1013 s−1 to the collision rate in the air at atmospheric pressure. This is consistent with the estimation above for the electron energy in the range from 0.1 to 10 eV.
3. PONDEROMOTIVE FORCE We can now retain the term m共v · ⵜ兲v in Eq. (1) neglected earlier. Furthermore, we relax the assumption that the laser field is spatially uniform over the transverse profile of the plasma filament and retain spatial derivatives of the laser pulse envelope. Specifically, for a linearly polarized laser pulse with a given amplitude of the first harmonic, E = E1共x,y,t − z/vg共兲兲exp关ik共兲z − it兴, exact expressions for the laser field have the form 1 EL = Exˆ + c.c., 2 1 kc BL =
2
Eyˆ −
1 ic 2
1e
ⵜ E ⫻ xˆ + c.c.
冋
ic
共9兲
册
vⴱ ⫻ − ⵜ E ⫻ xˆ + c.c. F = − m共vⴱ · ⵜ兲v + 4 4c 共10兲 Straightforward calculations reveal a well known result [22], which takes especially the simple form e2 F=−
4m2
共12兲
where a is a characteristic radius of the plasma filament and re = e2 / mc2 is the classical electron radius. In other words, the laser power in the SH should exceed 4.4 GW for the PF to be smaller than the RPF. The GW laser power is currently feasible, which provided the initial motivation for introducing into account the RPF and looking for a prospective physical mechanism that could create a dipole moment in the plasma filament [18]. It is worth noting, however, that the transverse part G⬜ of the RPF can produce larger emission from the plasma filament than that of the PF F⬜ even if the ordering Eq. (12) fails. Since G⬜ has a predominant direction across the entire section of the plasma filament, it produces a dipole polarization and, hence, a dipole radiation. On the contrary, F⬜ is directed mainly toward the center of the filament. It produces therefore a dipole radiation only due to the unavoidable ellipticity of the filament cross section. Ceteris paribus, a quadrupole radiation is less powerful and it is therefore usually neglected in comparison with the dipole radiation caused by the longitudinal part of the PF Fz [5].
4. SLOW MOTION
Similar formulas can be drawn for the SH. Approximate formulas (2) and (6) are obtained from Eq. (9) by dropping ⵜE and substituting kc with . The PF is identified as the sum of all terms with spatial derivatives of the pulse envelope, namely, 1
c共E22/8兲a2 Ⰷ 共c/2re兲mc2 ,
19
The combined force f = F + G causes the plasma electrons to execute relatively slow motion that can be characterized by the electron–ion separation vector . The latter is associated with the averaged velocity 具v典 = / t and obeys the equation
2 t
2
+␥
t
e =
f
m
E+
m
.
共13兲
It can be derived from Eq. (1) by a straightforward averaging procedure and by subsequently dropping the nonlinear term 共 / t , ⵜ兲 / t, which is justified by the smallness of in comparison with a characteristic radius a of the plasma filament. The polarization electric field E in Eq. (13) is related to the electron density perturbation ␦ne by the equation div E = 4e␦ne .
ⵜ 兩E兩 , 2
共11兲
if ␥ = 0. In plasma physics formula (11) is sometimes called the Miller force. The standard derivation of the Miller force relies on the Lagrangian coordinates where an observer follows individual fluid particles as they move through space (see, e.g., [23–25]). On the contrary, Eq. (10) gives the Miller force in the Eulerian coordinates that focus on specific locations in the space through which the fluid flows. The PF can be neglected if Fz Ⰶ Gz and Fy Ⰶ Gy. The first condition is equivalent to the inequality v2 / l Ⰶ evE / mc, where l = cL is the coherence length of the laser pulse with the time duration L. Putting here v ⬃ eE / m, one immediately concludes that the longitudinal PF Fz is small as compared with Gz if l Ⰷ c / ␥. The latter inequality is feasible even for short laser pulses. The condition Fy Ⰶ Gy leads to the inequality
In turn, ␦ne obeys the linearized continuity equation
␦ne t
冉 冊
+ div ne
t
= 0,
共14兲
which yields E = − 4ene .
共15兲
Putting Eq. (15) in Eq. (13) gives the final equation
2 t
2
+␥
t
+ p2 =
f m
,
共16兲
for , where p = 冑4e2ne / m is the plasma frequency, and ne is the local density of the electrons; in general, the density ne = ne共r , t兲 varies from point to point and also in time on a time scale different from the period 2 / p of the plasma oscillations.
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J. Opt. Soc. Am. B / Vol. 27, No. 1 / January 2010
Balakin et al.
The driving force f = f共r , t − z / vg兲 is assumed below to be a function of the radius vector r = 共x , y , z兲 and the time t⬘ = t − z / vg in a comoving frame of reference. Within a plasma filament, a small difference between the group velocities of the fundamental harmonic, vg共兲 ⬇ c共1 − p2 / 22兲, and that of the second one, vg共2兲 ⬇ c共1 − p2 / 82兲, can be neglected. Indeed, the mistiming length in the plasma, ⌬zp =
L 1/vg共2兲 − 1/vg共兲
⬇
82 3p2
cL ,
共17兲
3 p2 4 c
.
共18兲
In a neutral gas, ⌬kg =
2 c
关n − n2兴
共19兲
is substantially smaller being proportional to the gas pressure through the refraction indices of the gas, n and n2, at corresponding frequencies. Equation (18) is obtained from Eq. (19) by taking n = 冑1 − p2 / 2 and n2 = 冑1 − p2 / 共2兲2. Misphasing in the gas determines a relative phase difference between the first harmonic and the SH at the “entrance” to the plasma filament, whereas the misphasing in the plasma causes modulation of the driving force f within the filament. The modulation period, ᐉp = 2/兩⌬kp兩 = 8c/3p2 ,
共20兲
strongly depends on the density of the electron in the plasma filament. In what follows, the modulation is prescribed to the first argument r of the force f共r , t − z / c兲. The plasma misphasing length ᐉp substantially exceeds cL but can be as short as a few centimeters. Integrating Eq. (16) over time with the initial conditions = 0 and / t = 0 at t = −⬁ and a constant electron density gives
=
1 m冑p2 − ␥2/4 − 兲d .
冕
⬁
⬇ f共r,t − z/c兲/mp2 , within the laser pulse head, 兩t − z / vg兩 ⱗ L, where the force f assumes a maximal magnitude. Behind the head, t − z / vg Ⰷ L, the laser pulse excites a damping wake of standing electron plasma oscillations, and
⬇ − Im兵fp exp关− 共␥/2 + ip兲共t − z/c兲兴其/2mp ,
for the harmonic envelopes E1 and E2 to become separated in space is approximately 106 times greater than the laser pulse head length cL for a typical ordering ⬃ 103p. As to the mistiming of the harmonics envelopes in the gas, it must be minimized by a proper experimental setup to a negligible amount since the effect of the SH cannot be observed otherwise. For this reason, we substitute vg below with the speed of light in every occurrence of the retarded time, t − z / vg ⬇ t − z / c. On the contrary, in some circumstances the misphasing of the two harmonics should be taken into account. By misphasing we imply the phase factor exp关i2k共兲z − ik共2兲z兴 that enters the term E2 E2ⴱ in G⬜. Within the plasma filament, ⌬kp = 2k共兲 − k共2兲 ⬇ −
relax this assumption in Section 6. If force f varies slowly on the time scale p−1, i.e., pL Ⰷ , Eq. (21) reduces to
exp关− ␥/2兴sin关冑p2 − ␥2/4兴f共r,t − z/c
0
共21兲
The electron density can be approximated by constant in time behind the ionization front of the laser pulse. We will
where fp is the time Fourier transform, f⍀共r兲 =
冕
⬁
f共r, 兲ei⍀d ,
共22兲
−⬁
of the driving force at ⍀ = p. The current density j共r , t − z / c兲 = ene / t is calculated by differentiating Eq. (21) over time. Keeping only major terms yields j⬇
ene m
冕
⬁
e−␥/2 cos共p兲f共r,t − z/c − 兲d .
共23兲
0
5. ELECTROMAGNETIC EMISSION FROM THE PLASMA FILAMENT In a dipole approximation, the energy spectral intensity of the low frequency radiation generated by the slow motion of the plasma electrons per unit solid angle can be calculated directly from the well-known expression ⍀2
dE d⍀do
=
4c3
兩n ⫻ j⍀,K兩2 ,
共24兲
where n and do are, respectively, a unit vector and an element of the solid angle in the direction of radiation, K = n⍀ / c is the wave vector, ⍀ is the low frequency of the emitted radiation, and j⍀,K =
冕
dtd3rei⍀t−iKrj共r,t兲
is the space–time Fourier transform of the current density [see Eq. (66.9) in [26]]. Note that ⍀ is assumed to be positive in Eq. (24), where it is already taken into account that 兩j⍀,K兩 = 兩j−⍀,−K兩. Putting f共r,t − z/c − 兲 =
冕
d⍀ 2
f⍀共r兲e−i⍀共t−z/c−兲
in Eq. (23) and then integrating over , we obtain the time Fourier transform of the current density, e j⍀ =
2m
nef⍀共r兲ei⍀z/c
冋
1
1
␥/2 + ip − i⍀
+
␥/2 − ip − i⍀
册
.
When taking the spatial Fourier transform of the current, we assume the radius of the plasma filament a to be smaller than the emitted wavelength 2c / ⍀ so that
Balakin et al.
Vol. 27, No. 1 / January 2010 / J. Opt. Soc. Am. B
ne ⬇ Ne␦共x兲␦共y兲,
To proceed further, we approximate the dependency of the force f共r , t⬘兲 on the retarded time t⬘ = t − z / c in the head of the laser pulse by a tablelike peak with the duration L so that f共r , t⬘兲 = f共r兲H共t⬘ + L / 2兲H共L / 2 − t⬘兲, where H stands for the Heaviside step function. Then,
where Ne = 兰兰nedxdy is the linear density of the electrons in the filament and ␦ stands for a delta function. In this approximation, only the driving force averaged over the filament cross section, 具f典 =
冕冕
具f⍀典 =
dxdynef/Ne,
冕冕
具f⍀典 = 具f共r兲典L sinc关⍀L/2兴,
dxdynef⍀/Ne ,
eNe 2m
冕
具f⍀典ei共⍀/c−Kz兲zdz 1
+
␥/2 − ip − i⍀
册
冋
1
␥/2 + ip − i⍀
具f共r兲典 = f0H共z + L/2兲H共L/2 − z兲cos关 + ⌬kpz兴,
d⍀do
=
⍀2 4c3
冉
eNeLL 2m
冋 冉
+ sinc
冊
共␥2/4 + ⍀2兲sinc2共⍀L/2兲
2
共␥2/4 + 共p − ⍀兲2兲共␥2/4 + 共p + ⍀兲2兲
冊 冉
再冋 冉 sinc
⌬kpL ⍀L ⌬kpL sin2 + sin2 − − sinc c 2 2 c 2 2
⍀L
共27兲
where f0 is a constant vector, is the phase shift between the two harmonics due to optical path difference in the gas, and ⌬kpz is the phase modulation within the plasma filament. Both and ⌬kp should be set to zero for the PF F and the longitudinal part of the RPF Gz. Performing the integration in Eq. (25) and putting the result in Eq. (24) gives an expression that is a bit cumbersome expression,
共25兲
,
does not depend on the transverse components Kx , Ky of the wave vector. The longitudinal component of the wave vector Kz is related to the angle between K and the axis z of the plasma filament by the equation Kz = 共⍀ / c兲cos so that ⍀ / c − Kz = 2共⍀ / c兲sin2 / 2.
dE
共26兲
where sinc共兲 = sin共兲 / . We also take into account that the plasma filament has a finite length L, and some components of the driving force (namely, G⬜) can be modulated due to the effect of the misphasing as discussed above. We simulate both these facts by writing
determines radiated power and the Fourier transform of the current density, j⍀,K =
21
冊册
冊 冉
⌬kpL ⍀L ⌬kpL sin2 + sin2 − + sinc c 2 2 c 2 2
⍀L
2
冎
sin2 兩n ⫻ f0兩2 .
冊册
2
cos2 共28兲
It simplifies to dE d⍀do
=
⍀2 4c
3
冉
eNeLL m
冊
共␥2/4 + ⍀2兲sinc2共⍀L/2兲
2
共␥ /4 + 共p − ⍀兲 兲共␥ /4 + 共p + ⍀兲 兲 2
2
if ⌬kpL / 2 Ⰶ 1, that is, if the plasma filament length is substantially smaller than the modulation length Eq. (20), L Ⰶ ᐉ p. This constitutes our principal result describing the spectral and angular distribution of the radiation. The frequency spectrum has a maximum at the plasma frequency, corresponding to p / 2 ⬃ 1 THz for ne = 1016 cm−3. Experimental data exhibit a broadband frequency spectrum rather than a resonantlike one. It would mean either that the friction coefficient ␥ is comparable with p or that the variation p across the plasma filament cross section should be taken into account. This can be done by averaging f / 关␥ / 2 ± ip − i⍀兴 in Eq. (25) rather than just the force f alone. In our opinion, both these effects are equally important. Although wide, the frequency spectrum in most experiments with femtosecond lasers is narrower than one could evaluate from the spectral width / L of the seed laser pulse. It means that the exact shape of the seed pulse is not very important, and the factor sinc2共⍀L / 2兲 can be dropped from Eqs. (28) and (29) as it is approximately
2
2
sinc2
冉
⍀L c
sin2
2
冊
cos2 兩n ⫻ f0兩2 ,
共29兲
equal to 1 within the main part of the spectrum. We utilize this observation in Section 6. As it is seen from Eqs. (28) and (29), the emitted terahertz power is affected by the phase difference between the fundamental and the SH pulses caused by the dispersion of the refractive index along the path of these optical pulses in the gas to the foci, as reported by Cook and Hochstrasser [7] and later by Kress et al. [9] and Chen et al. [27]. Furthermore, Eq. (28) also describes the effect of the dispersion inside the plasma filament, and the latter limits the effective length of the filament that contributes to the terahertz emission. In particular, Eq. (29) predicts that the total emitted power oscillates from zero to a maximal value as the phase shift varies with the gas pressure. The more accurate Eq. (28) excludes lowering the emitted power down to zero as the result of the phase modulation within the plasma filament. Instead of being zero, the total radiated power falls down to a minimal value, which depends on the ratio of the plasma filament length L to the modulation length ᐉp. Since = 0 both for the PF and longitudinal RPF, detecting the power oscilla-
22
J. Opt. Soc. Am. B / Vol. 27, No. 1 / January 2010
Balakin et al.
tions would certainly indicate that the transverse RPF plays a dominating role in the emission of the terahertz waves from the plasma filament. It is remarkable that the maximal emitted power is proportional to the square of the total number NeL of the free electrons in the plasma filament. The angular distribution also provides crucial information regarding which of the two forces plays a dominating role. According to Eq. (29), the angular distribution contains multiple lobes, the angular positions of which are defined by the condition 2L sin2 / 2 = N for the emitted wavelength = 2c / ⍀ and integer number N = 0 , 1 , 2 , . . .. The lobe N = 0 does not exist if the force f0 is directed along the plasma filament. This occurs when the longitudinal force is greater than the transverse one, i.e., fz Ⰷ f⬜. In this case, the angular distribution is proportional to sin2 sinc2共⍀L sin2共 / 2兲 / c兲, and the main lobe, N = 0, is empty inside resembling an inverse Cherenkov cone as shown in Fig. 1(a) for the case L / = 50. The lobe N = 1 is the strongest and corresponds to the cone opening angle
⬇ 冑2/L.
共30兲
On the contrary, if the transverse force is dominant, i.e., f⬜ Ⰷ fz, the angular distribution is proportional to 共1 − sin2 ␣ sin2兲sinc2共⍀L sin2共 / 2兲 / c兲, where ␣ is the azimuthal angle (around the filament axis z) counted from the direction f⬜. In this case, the main lobe is filled. It is also slightly distorted in the azimuthal direction but this distortion is very weak since 共1 − sin2 ␣ sin2 兲 ⬇ 1 for small . In addition, the contribution of the longitudinal force is suppressed by the small factor sin2 ⬃ 2 / L. It means that the domination of the transverse force over the longitudinal one requires a less severe condition f⬜ Ⰷ fz共 / L兲 to be satisfied instead of the inequality f⬜ Ⰷ fz specified above from superficial consideration. Comparing Figs. 1(a) and 1(b) shows that measuring the angular distribution of the radiated power allows determining the dominant direction of the driving force f. However, it is important to note that the contribution of the transverse PF F⬜ to the emitted power is substantially suppressed and might seem to be lower than that of the RPF G⬜ of the same magnitude. It follows from the fact that 兰兰neⵜ⬜兩E兩2dxdy = 0 for an axisymmetric distribution of both ne and 兩E兩2. The angular distribution becomes even more complicated if ᐉp ⬃ L, when the exact formula (28) should be used instead of Eq. (29). In this case, not only the total
emitted power varies with the gas pressure (through the phase shift ) but also the angular distribution. A more certain conclusion that follows from Eq. (28) is that it predicts a drastic decrease in the emitted power when the plasma filament length greatly exceeds the modulation length, L Ⰷ ᐉp. In this case, the terms in the curly brackets drop as 共ᐉp / L兲2, and the total power occurs to be proportional to 共Neᐉp兲2 instead of 共NeL兲2. Thus, ᐉp is an effective length of the plasma filament that effectively contributes to the radiation.
6. EFFECT OF THE TRANSIENT PHOTOCURRENT We now relax the assumption that photoelectrons are born with zero average velocity. Although the MPI is essentially a quantum process, we employ a simple classical model in the hope that it catches such a crucial feature of the photoionization as the dependency on the phase shift between the first harmonic and the SH of the laser field. Let us consider the motion of a free electron in the twocolor laser field, E共t兲 = E cos共t兲 + E2 cos共2t + 兲.
We assume that the electron is born in the result of the MPI at the instant of time tN and that its velocity at that moment is equal to zero. Then, we have v共t兲 = v sin共t兲 + ␣v sin共2t + 兲 + vN ,
120
75
60
135
120 45
150
105 90
75
vN = − v sin共tN兲 − ␣v sin共2tN + 兲.
30
45
150
30
165
15
165
15
180
0
180
0
195
345 195
210
330
225
(a)
240
255 270 285
300
345
210
315
330
225
(b)
315 240
255 270 285
共33兲
Employing the classical model of the photocurrent generation [16,17], we make one more proposition, the validity of which we will discuss later. Namely, we assume that due to the extremely large nonlinearity of the MPI process tearing-off a bound electron from an atom occurs exclusively at the instants of time, corresponding to the maxima of the absolute magnitude of the electric field strength. Making use of the smallness of the parameter ␣ = E2 / E in realistic experimental conditions, we find the phase of the fundamental harmonic at these instants of time, 共34兲
where N stands for an integer. Having calculated the average velocity of an electron,
60
135
共32兲
for t ⬎ tN, where v = eE / m and ␣ = E2 / E. The last term in Eq. (32), which does not depend on time, is set by the condition v共tN兲 = 0 and turns out to be the average velocity of the electron in the laser field,
tN = N − 2␣ sin cos N, 105 90
共31兲
300
Fig. 1. (Color online) Angular distribution of the terahertz radiation computed by Eq. (29) for L / = 50: (a) driving force is directed along the plasma filament; (b) driving force is perpendicular to the plasma filament.
3
vN = 2 ␣v sin ,
共35兲
born at these moments, we note that it does not depend on N and, hence, is the same both for odd and even maxima. However, these maxima have different magnitudes, EN ⬅ 兩E共tN兲兩 = 兩E兩关1 + ␣ cos N兴.
共36兲
The numbers of the electrons produced at odd and even maxima are different.
Balakin et al.
Vol. 27, No. 1 / January 2010 / J. Opt. Soc. Am. B
Generally speaking, this difference might be very huge even for a very small ratio ␣ since a photon with doubled frequency has doubled energy and, thus, a two times smaller number of such photons is required to complete the act of ionization. However, for the sake of simplicity we assume that the parameter ␣ is so small that the photoionization is produced exclusively by the fundamental harmonic of the laser field. It is then reasonable to think that the number of the electrons produced in the odd maxima differs from that produced in the even ones by the value of the order of ␣. If it is not so and the ionization by the SH dominates over the fundamental harmonic, one needs to think that the electrons are predominantly born at the maxima of the SH but we will not consider this case. Since the number of the electrons produced at different moments differs by a small amount of the order of ␣ according to our assumption, the difference can be neglected when calculating the total (macroscopic) initial momentum of the electrons because their average velocity Eq. (35) itself contains the small parameter ␣. We note also that any initial spread of the electron velocities does not appear in our model. It means that a hydrodynamic approximation with zero temperature can be used to describe further the motion of the electron fluid. To include the effect of the initial electron velocity, we extend the equations of slow motion from Section 4. First, we need to add a source producing n˙ electrons per unit of time in a unit of volume and take into account that the electrons acquire an initial velocity v0 when they are born. Respectively, we add the term n˙ to the RHS of the continuity equation,
ne t
共37兲
+ div共nev兲 = n˙ ,
and the summand n˙v0 to the RHS of the equation of motion,
t
nev + ⵜ · 共nevv兲 =
ene m
E − ␥ n ev + n e
f m
+ n˙v0 .
Recall that E stands here for a slow polarization field appearing due to the spatial separation of the electrons from the neutralizing ion background. Excluding ne / t from the second equation with the aid of the first one, we obtain
t
v + 共v · ⵜ兲v =
e m
E − ␥v +
n˙
f m
+
ne
关v0 − v兴.
共38兲
To proceed further, we linearize Eqs. (37) and (38), assuming the velocity v and the perturbation of the electron density ␦ne to be equally small quantities. By an unperturbed electron density we shall imply the quantity n=
冕
t
n˙共t兲dt,
共39兲
−⬁
which is the formal solution of the continuity equation for v ⬅ 0. In a focus of a femtosecond laser pulse, secondary ionization has no time to enter the scene and the source of the electrons n˙ can therefore be considered as a given function of time. Taking this fact into account when lin-
23
earizing Eqs. (37) and (38), putting there ne = n + ␦ne, and expressing the velocity v = / t through the vector , we obtain
冉 冊
␦ne
+ div ne
t 2 t
2
+␥
t
e =
m
t
n˙
f E+
m
+
n
共40兲
= 0,
冋 册 v0 −
t
.
共41兲
Since Eq. (40) exactly coincides with Eq. (14) from Section 4, expression (15) for the polarization electric field E also remains valid. Putting it in Eq. (41) gives the final equation
2 t
2
+␥
t
+ p2 =
n˙
f m
+
n
冋 册 v0 −
t
,
共42兲
for , where p = 冑4e2n / m is again the plasma frequency. This equation differs from Eq. (16) by the last term in the RHS. The driving force f = f共r , t − z / vg兲 again can be interpreted as a function of the radius vector r = 共x , y , z兲 and the time t⬘ = t − z / vg in a comoving frame of reference. The same assumption is also applicable to the quantities n and v0. Integrating Eq. (42) over time should be done with the initial conditions = 0 and / t = 0 at t = −⬁. Since the plasma density n, the plasma frequency p, the driving force f, and the initial velocity v0 depend on time, the integration can be generally done only numerically. There is, however, a practically important case, when one can present an approximate solution. As has been noted at the end of Section 4, the period 2 / p of the plasma oscillations in present-day experiments with femtosecond lasers does not exceed the duration of the laser pulse. Restricting ourselves to the solution of Eq. (42) at larger times, we can say that the separation vector in an arbitrary point with the coordinate z starts to differ from zero only after the laser pulse head passes through the point, i.e., at t ⬎ z / vg. Behind the pulse head the unperturbed electron density remains constant as well as the plasma frequency. As a result, Eq. (42) is transformed into the ordinary differential equation
2 t
2
+␥
t
+ p2 = 0,
共43兲
with zero RHS and constant coefficients, which depend on the point coordinates only as on the parameters. This equation should be solved with the initial conditions
t
1 =
n
冕
= 0,
共44兲
关nf/m + n˙v0兴dt,
共45兲
at t = z / vg + 0. It is assumed here that n before the integral sign in the RHS of Eq. (45) is the electron density established after the laser pulse head passing, and the range of the integrations over the time comprises the entire interval of the pulse passing (e.g., from t = z / vg − L to t = z / vg
24
J. Opt. Soc. Am. B / Vol. 27, No. 1 / January 2010
Balakin et al.
+ L). The solution of the reduced problem [Eqs. (43)–(45)] is trivial,
=
e−␥共t−z/vg兲/2
冑p2 − ␥2/4
冋冑 冉 冊册冏 冏
sin
p2 −
␥2 4
t−
z
vg
t
. t=z/vg+0
共46兲 Elementary comparison of the summands in the integrand in Eq. (45) shows that the effect of the initial electron momentum acquired by the electrons at the act of the ionization causes a stronger plasma polarization if the momentum exceeds the momentum gained by the electrons from the driving force for the entire duration of the laser pulse, i.e., mv0 Ⰷ fL. A formal estimation v0 ⬃ ␣v, following from the classical treatment in the beginning of this section, leads to the conclusion that the condition mv0 Ⰷ fL is satisfied practically for any laser power feasible to the date. However, available quantum theories predict a substantially lower value of v0, especially in the case of the laser pulse with the duration significantly exceeding the period of the laser field, L Ⰷ 1. Indeed, the notion of the instant of time of the ionization has no evident sense in quantum physics. According to the uncertainty principle, the instant of time of the ionization is determined at most with the accuracy of the order of the period of the bound electron gyration around the atomic core 2 / a (if the frequency a = J / ប, corresponding to the ionization potential J can be interpreted so) or with the accuracy of the ionizing field period 2 / . As a quantum theory of the photoionization [28,29] says, the ionization occurs at times close to the absolute maximum of the laser field (taking into account the field envelope time profile) in a so called limit of low frequency, when v Ⰷ ve = 冑2J / m [30]. According to the results of [29], the velocity distribution of the produced electrons has a maximum near vmax = 兰0⬁dteE共t兲 / m. Although the computation in [29] was performed for a special case, where E共 −t兲 = E共t兲, which corresponds to the phase shift , multiple of , its result assumes that vmax drops quickly as the pulse duration increases since the integral 兰0⬁dteE共t兲 / m decreases in proportion to 1 / L for a pulse with a tablelike time envelope and even more quickly for smoother envelope profiles. For example, it gives vmax ⬀ exp共−1 / 22L兲 for the Gaussian profile. For a monochromatic wave, the theory yields vmax = 0. On the other hand, a numerical solution of the Schrödinger and experimental data, reported in [31], supports the dependency of the average velocity on the phase shift proportional to sin , as predicted by our simple classical model. However, the same result contradicts to the results of [29] as the latter gives nonzero vmax for = N.
7. EXPERIMENTAL RESULTS AND DISCUSSION Experimental evidence is needed to establish the veracity of the model presented above. We used an experimental scheme analogous to that described in [7]. We employed an amplified Ti:sapphire laser system providing fundamental 120 fs and 800 nm pulses at the
repetition rate of 1 kHz. The SH is generated in a beta barium borate crystal (BBO, type-I, 300 m thick). Both the intense fundamental laser pulses and their SH descendants are focused into a gas cell by a lens with a focal length of f = 18 cm. Radiated terahertz waves are collected by a set of crystal quartz lenses and then focused by an off-axis parabola 共f = 25 mm兲 onto 300 m thick 具110典 oriented ZnTe crystal for electro-optical detection. The metal gas cell of 100 mm length has a window with a diameter of 25 mm. The front window is made of a 300 m thick fused quartz plate and a rear window is fabricated from Teflon. The gas cell is filled with air or with a noble gas (argon, neon, xenon, or krypton) at adjustable pressure in the range from 0 to 1.5 bar. Pure gas can be pumped into or evacuated from the cell trough a gas diluting system. To separate and reduce the contribution of the terahertz wave from the BBO crystal, we used an adjustable special filter placed inside the gas cell in the focus area. According to basic principles of the terahertz time domain measurements, the temporal profile of the emitted terahertz wave is measured by scanning the time delay of a probe beam at the fundamental frequency relative to the terahertz pulse. The phase shift between the fundamental and the SH pulses can be changed [7,9,11] by varying the distance ⌬l between the SH generation BBO crystal and the laser focus. In addition, varies with the gas pressure in the cell since both n − 1 and n2 − 1 are proportional to the pressure but with different coefficients of proportionality. In both cases, the modulation of the terahertz wave amplitude has been detected in accordance with Eq. (28). Similar behavior has also been observed by Kress et al. [9] who moved a SH nonlinear crystal (BBO) away from to the focus. In this case the phase shift can be written as = 共2 / c兲共n − n兲⌬l. The variation in the phase shift leads also to the changes in the temporal profile of the terahertz field and its polarity [11,16] in agreement with our discussion at the end of Section 5. In our experiments, the phase shift changes by as the gas (Xe) pressure changes by an amount in the range from 0.6 to 0.7 bar. Figure 2 illus.
.
.
.
.
.
.
.
.
.
.
.
.
.
Fig. 2. Dependence of the terahertz field amplitude generated in the Xe versus the gas pressure. In the figure, open squares correspond to 980 mW and the dark triangles correspond to the 600 mW of the overage power of the fundamental field.
Balakin et al.
trates this statement. Using indices of refraction from [32–35], we have estimated the pressure periods to be 1.1, 0.6, 0.25, and 10 atm for Ar, Kr, Xe, and Ne, respectively. These estimates agree fairly well with experimentally observed values, especially at lower pressures.
Vol. 27, No. 1 / January 2010 / J. Opt. Soc. Am. B
25
aspects of the laser-induced terahertz radiation, and M. Nazarov for the assistance in the experiments. This work was supported by the Russian Foundation for Basic Research (RFBR) in the frameworks of grants RFBR 08-0200869 and 09-02-12198 and by the Russian Federal Agency for Science and Innovation (Rosnauka), the state contract 02.740.11.0223.
8. CONCLUSION In this paper we discussed basic principles of the terahertz wave generation initiated by the optical breakdown in air of a noble gas. We focused mainly on the two-color scheme of the terahertz wave generation. We presented a simple classical model of the multiple photoionization in a two-color laser field that hopefully catches some peculiar features of the process. In particular, we emphasize that the contribution of the radiation pressure force (RPF) to the emitted terahertz power varies with the phase shift due to the optical phase difference in the air to the laser focus in proportion to cos , whereas the contribution of the transient photocurrent is proportional to sin . We have distinguished the RPF from the ponderomotive force (PF) and, for the first time, we have calculated the transverse component of the RPF. We have shown that the effect of the second harmonic (SH) is stronger when it is polarized differently from the first harmonic. In addition, the effect of the transverse PF is suppressed as compared with the effect of the transverse RPF because the PF varies across the section of the laser beam. Furthermore, the effect of the longitudinal driving force is suppressed as compared with the transverse force by the factor / L. Both empty and filled angular diagrams of the terahertz emission (see Fig. 1) are allowed by our theory and both types of the diagram were observed experimentally although no special care has been taken so far to quantitatively validate experimental data against our theory (as it was not available earlier). By inspecting the angular diagram of the terahertz radiation, one can deduce which component—the longitudinal or the transverse—makes a dominating contribution to the emitted terahertz power. We showed that the effective length ᐉp [given by Eq. (20)] of the emitting part of the plasma filament is related to the angle spread of the terahertz radiation angular diagram if ᐉp Ⰶ L. We explained that the effective length shortens due to the optical dispersion within the filament in inverse proportion to the electron density. We attributed the broadband character of the terahertz radiation to the spatial inhomogeneity of the electron density within the plasma filament. If we compare our theoretical speculations with the complete experimental data available so far, we conclude that they do not allow validating our theory without doubts since some experimental results are mutually exclusive. We hope that future experiments will resolve this discrepancy.
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18.
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31. 32.
33. 34. 35.
V. S. Popov, “Multiphoton ionization of atoms by an ultrashort laser pulse,” JETP Lett. 73, 1–5 (2001). This condition can be rewritten as cE2 / 8 3 8 3 3 2 Ⰷ 共␣ mc / 32re 兲共J / JH兲 / 共J / ប兲 , where ␣ is the fine structure constant, re is the classical electron radius, J / JH is the ratio of the potential of ionization to that of the atom of hydrogen, and J / ប is the number of photons required for the ionization. Numerically, this condition means that the laser power flux density should exceed 8.8 ⫻ 1015共J / JH兲3 / 共J / ប兲2 W / cm2. N. Karpowicz and X.-C. Zhang, “Coherent terahertz echo of tunnel ionization in gases,” Phys. Rev. Lett. 102, 093001 (2009). C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of krypton and xenon and their relation to those of helium and argon,” Proc. R. Soc. London, Ser. A 84, 2805–2807 (1910). C. Cuthbertson and M. Cuthbertson, “On the refraction and dispersion of neon,” Proc. R. Soc. London, Ser. A 83, 149–151 (1910). S. A. Korff and G. Breit, “Optical dispersion,” Rev. Mod. Phys. 4, 471–503 (1932). E. R. Peck and D. J. Fisher, “Dispersion of argon,” J. Opt. Soc. Am. A 54, 1362–1364 (1964).
