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Resonant X-ray second harmonic generation in atomic
gases
Ji-Cai Liu, Catalin Miron, Hans Ågren, Sergey Polyutov, Faris Gel’Mukhanov
To cite this version:
Ji-Cai Liu, Catalin Miron, Hans Ågren, Sergey Polyutov, Faris Gel’Mukhanov. Resonant X-ray second harmonic generation in atomic gases. Physical Review A, American Physical Society 2019, 100 (6), �10.1103/PhysRevA.100.063403�. �hal-02442214�
Resonant X-ray second harmonic generation in atomic gases
Ji-Cai Liu1,2∗, Catalin Miron3,4,5, Hans ˚Agren6,7,
Sergei Polyutov8,9, and Faris Gel’mukhanov6,8,9,10
1Department of Mathematics and Physics,
North China Electric Power University, 102206 Beijing, China
2Collaborative Innovation Center of Light Manipulations and Applications,
Shandong Normal University, Jinan 250358, China
3Synchrotron SOLEIL,
l’Orme des Merisiers, Saint-Aubin, BP 48,
91192 Gif-sur-Yvette Cedex, France
4Extreme Light Infrastructure Nuclear Physics (ELI-NP),
‘Horia Hulubei’ National Institute for Physics and Nuclear Engineering,
30 Reactorului Street, RO-077125 Mˇagurele,
Judet Ilfov, Romania
5LIDYL, CEA, CNRS,
Universit´e Paris-Saclay, CEA Saclay,
91191 Gif-sur-Yvette, France
6Theoretical Chemistry & Biology,
School of Biotechnology, Royal Institute of Technology,
S-106 91 Stockholm, Sweden
7Department of Physics and Astronomy,
UppsalaUniversity, SE-75120Uppsala, Sweden
8Siberian Federal University,
660041 Krasnoyarsk, Russia
9 Kirensky Institute of Physics,
Federal Research Center KSC SB RAS, 660036 Krasnoyarsk, Russia
10Institute for Methods and Instrumentation
in Synchrotron Radiation Research FG-ISRR,
Helmholtz-Zentrum Berlin f¨ur Materialien und Energie Albert-Einstein-Strasse 15,
12489 Berlin, Germany
Abstract
We explore the X-ray second harmonic generation process induced by resonant two-photon ab-sorption in systems with inversion symmetry. We show that this process becomes allowed in the X-ray region due to non-dipole contributions. It is found that while a plane wave pump field generates only a longitudinal second harmonic field, a Gaussian pump beam creates also a radially polarized transverse second harmonic field which is stronger than the longitudinal one. Contrary to the longitudinal component, the transverse second harmonic field with zero intensity on the axis of the pump beam can run in free space. Our theory is applied to Ar and Ne atomic vapours and predicts an energy conversion efficiency of X-ray SH generation of 3.2 × 10−11 and 1.3 × 10−12, respectively.
I. INTRODUCTION
Modern X-ray Free Electron Laser (XFEL) facilities can deliver high intensities 1015−1019
W/cm2 making it possible to significantly populate a core-excited state and even create
population inversion and X-ray lasing [1–7]. At these intensities, X-ray matter interaction becomes nonlinear creating room for studies of nonlinear effects like stimulated X-ray Raman scattering [3–5, 8–10], pulse compression [3–5], X-ray four-wave mixing [3–5, 11, 12], and nonlinear wave mixing of X-ray and near-infrared beams [13]. Special attention was paid to the competition between stimulated X-ray emission and Auger decay [14–16]. Second harmonic generation (SHG) is a nonlinear optical process of sum frequency generation which produces new photons with twice the frequency. SHG has traditionally been studied as an even-order nonlinear optical effect allowed in media without inversion symmetry[17] and is one of the best-understood nonlinear effects in optics[18]. In light of the XFEL development its study in the Angstrom regime, e.g., on the natural scale of atomic and molecular structure of matter, has become of great interest both from a fundamental and practical viewpoint. A pioneering study by Shwartz, Yudovich et al[19, 20] gave recently experimental evidence for ”off-resonant” SHG in diamond in the hard X-ray region with a pump frequency of 7.3 keV.
In the present work we show that due to the large momentum of the photon k the nonlinearity in the X-ray region is different from conventional nonlinearities in the visible regime and that SHG is generally possible to observe for centro-symmetric systems even when phase matching conditions do not prevail. We present a theoretical study of X-ray SHG in atomic gases induced by resonant two-photon absorption (TPA). We show that the plane wave pump field can create only longitudinally polarised X-ray second harmonic (SH) fields which can not propagate in free space (see however ref.[21]), but also that a Gaussian pump pulse induces in addition transverse second harmonic fields which contrary to the longitudinal component can run in free space. Our idea is, in a certain sense, inverse to the use of the longitudinal component of focused light beams in laser particle accelerators[22]. Another important feature of the SHG problem studied here is that the transverse field, being strictly equal to zero on the beam axis, has an unusual radial polarization.
Our work is organized as follows. We outline in Sec. II A the basic theory of SHG
the following Sec. II B we show that a Gaussian pump field creates also the transversely polarized SH field. Sec. II C is devoted to the analysis of the longitudinally and radially polarized SH fields. We shed light on the role of the absorption of an SH field in Sec. II D.
Some theoretical details can be found in Appendixes A, B and C. We discuss our results
further in Sec. III where we numerically analyze the efficiency of SHG in Ne and Ar vapours. Finally, in Sec. IV, we come to the conclusions.
II. THEORY
Quantum mechanically, the second order nonlinearity in the optical susceptibility
origi-nates from a perturbational solution of the Schr¨odinger’s equation. To get insight into the
physics of the SHG process in the X-ray region, we consider the propagation of an X-ray
pump field Ep in an atomic gas far away from the absorption edge. To induce the SHG
we choose twice the frequency of the pump field to be resonant with the frequency of a
two-photon transition 2ω ≈ ω10. The scheme of SHG is shown in Fig.1 where the pump
field resonantly promotes the 1s electron of atom to the np unoccupied level. The resonant population of the state |1i in the course of TPA is followed by the emission of the SH field E. Let us start from the atom-field interaction which reads as (we use SI units)
V = V(1)+ V(2) = − e 2mc(p · Ap+ Ap · p) + e2 2mc2A 2 p, (1)
where m and e are the mass and charge of the electron, respectively. c is the speed of light and p is the operator of electronic momentum. Below we will use more frequently the electric field instead of the vector potential, E = −∂A/∂t. The square of the vector potential of the
pump field A2p describes the TPA process in the first order of perturbation theory while the
scalar product p · Ap contributes to the TPA in second order of perturbation theory.
A. Plane wave pump field
It is instructive to consider first the interaction with the simplest and most fundamental
electromagnetic wave, the transverse plane wave Ap = (Ap/2) exp(−ı(ωt−k·r−k·r(e)))+c.c
Ep = 1 2Epe −ı(ωt−k·r−k·r(e)) + c.c., Ep = eEp(0) = − ∂Ap ∂t = ıωAp (2)
with the polarization e being orthogonal to the photon momentum k. Here r(e) is the
coordinate of the electron with respect to the atom and r is the radius vector of atom in the laboratory frame. To avoid unnecessary complexity (see also below) we will focus only on
the A2p term assuming that the wave length of the photon is longer than the size of the core
orbital ka 1 V(2) = e 2 8mc2A 2e−ı2(ωt−k·r(e)−k·r) + c.c. + const (3) ≈ e 2 8mc2A (0)2 p e −ı2(ωt−k·r) (1 + ı2k · r(e)) + c.c. + const.
The term A(0)2p being independent on the electron radius vector r(e)can not induce transitions
between electronic states. Thus the transition between the ground (s) and core-excited (p) states is induced solely by the the matrix element
V10(2) ≈ −ı eE
(0)2 p
4mc2ω2(k · d10)e
−ı[(2ω−ω10)t−2k·r] (4)
of the second term k · r(e)on the right-hand side of eq.(3). The rotating-wave approximation
is used here by keeping only the near resonant term. This pure non-dipole process opens
the s → p TPA channel with the transition dipole moment d01 = er01 = eh0|r(e)|1i (Fig.1).
We chose the axis z of quantization to lie along the photon momentum k. In this frame
the pump field populates only the npz level (see Fig.1) and the problem is reduced to the
interaction with a two-level atom with the transition dipole moment parallel to the photon momentum
d01 k k. (5)
The resonant TPA population of the core-excited state of p-symmetry is followed by the dipole allowed one-photon transition p → s which creates the SHG field with the double frequency 2ω. This explains why the SHG is possible in systems with inversion symmetry in the X-ray region.
To quantify the studied process one should compute the polarization P. The induced macroscopic polarisation of the medium being the expectation value of the dipole moment d is specified in terms of the density matrix ρ(t)
P = N Tr(dρ) = N (d01(t)ρ10(t) + d10(t)ρ01(t)) , (6)
where N is the concentration of atoms and d01(t) = d01exp(ıω10t) is the dipole moment
ρ10(t) = %10exp(−ı(νt − 2k · r)) satisfies the following kinetic equation in the interaction picture[23] ∂ ∂t + Γ − ıν %10= 1 ~m eEp(0) 2ω 2 (k · r10) (ρ11− ρ00) , (7)
where ν = 2ω − ω10 is the detuning from the two-photon resonance and Γ is the lifetime
broadening of core-excietd state |1i. We neglect the very weak depopulation of the ground
state in the course of the two-photon absorption, (ρ00 ≈ 1, ρ11 1) and assume that the
duration τ of the pump pulse is longer than the lifetime of the core excited state 1/Γ. In this case one can use the stationary solution of eq.(7)
%10= %∗01 = − 1 ~m eEp(0) 2ω !2 (k · r10) Γ − ıν (8)
to find the induced macroscopic polarization taking into account eq.(5)
P = P e−ı2(ωt−k·r)+ c.c., (9) P = ˆkp, p = − eE (0) p 2ω !2 N ker201 m~(Γ − ıν).
Here ˆk = k/k is the unit vector along k. One can see that the pump radiation creates a
macroscopic polarization P oriented along the direction of propagation of the pump field k, and hence the SH field E, which is created in the course of the spontaneous transition |1i → |0i, is also parallel to k. This longitudinal field exists everywhere where there is pump field and the medium and this field copies exactly the polarization according to Maxwell’s
equation for the induction, ∇ · D = ∂(0E + P)/∂z = 0:
E = −2
0
P 6= 0, D = 0, H = 0. (10)
This does not contradict the well known fact that the plane wave longitudinal field does not exist in free space[21]. This statement means that the longitudinal field can not propagate in free space. The longitudinal field E exists only in the region where the pump field creates longitudinal polarization P ∝ k. This longitudinal field oscillating in time and space is a pure electric field, H = 0.
As we have already noticed above the TPA process is a first-order process with respect
to A2p and a second-order process with respect to p · Ap. Here we study the two-photon
result in the same orientation of the TPA induced polarization we consider here only the
A2
p contribution. The taking into account of the p · Ap TPA process will result only in a
rescaling of the SHG efficiency.
B. Gaussian pump beam and paraxial equation
In this section we will show that a Gaussian pump beam
Ep = 1 2Epe −ı(ωt−kz)+ c.c., (11) Ep = ˆxEp(0)g t − z c w0 w(z)exp − % 2 w2(z) exp ı k % 2 2R(z) − ψ(z)
makes it possible to transform the longitudinal SHG X-ray field into a transfer field which
can propagate in free space in contrast to the pure longitudinal field. Eq. (11) for pulsed
Gaussian beam is obtained in Appendix A by convoluting of the fundamental Gaussian mode with the Gaussian distribution of spectral components. As shown in Appendix A, Ep satisfies
paraxial equation. Eq.(11) identifies R(z) = z (1 + (zR/z)2) as the radius of curvature of the
wavefront of the beam at z, w0 as the beam waist and g(t) = exp(−t2/2τ2) as the temporal
shape of the pulse with duration τ . Here w(z) = w0p1 + (z/zR)2, ψ(z) = arctan(z/zR),
ρ =px2+ y2, w
0/zR∼ 1/kw0 1. The Gaussian beam remains well collimated up to the
Rayleigh range zR = kw20/2 (Fig.2).
Since the wavefront is not orthogonal to z, as one can see from the phase φ = 2k(z+ρ2/2R)
of Ep2 ∝ exp(ıφ), the polarization P is slightly tilted from the z-axis. To find the matrix
element V01(2) of the interaction with the Gaussian pump beam (11) we need the value of this
interaction at the point of the electron r(e) with respect to the atom r = (ρ, z), namely at
r + r(e),
h0|eı(φ+δφ)|1i ≈ eıφh0|1 + ıδφ|1i = κ · h0|r(e)|1ieıφ ı2k, (12)
where we used the Taylor expansion φ(r+r(e)) = φ(r)+δφ with δφ = ∇φ·r(e). Similar to the
derivation of eq.(9) one obtains a polarisation that is oriented along ∇φ ≡ (∂zφ, ∂ρφ) = 2kκ
P = −κ eEp 2ω 2 N ker2 01 m~(Γ − ıν), κ = ˆz + ˆρ ρ R (13)
instead of the beam axis ˆz k k.
Let us write the optical wave equation for the SHG field E
∇(∇ · E) − ∇2E + 1 c2 ∂2E ∂t2 = −µ0 ∂2P ∂t2 (14)
in the usual manner[18] starting from the couple of Maxwell’s equations (in SI units) for nonmagnetic materials (µ=1) ∇ × E = −µ0 ∂H ∂t , ∇ × H = ∂D ∂t . (15)
Contrary to conventional theories[18] where ∇ · E ∼ (k · e) = 0 for the transverse electro-magnetic field (k ⊥ e), we can not ignore ∇ · E here. This is because the polarization P is essentially a longitudinal one (see eq.(13)): ∇ · P 6= 0. To resolve this problem we use the
Maxwell’s equation for the induction D = ε0E + P
∇ · D = 0, ∇ · E = −1
0
∇ · P, (16)
which makes it possible to rewrite the wave equation (14) as follows
−∇2E + 1 c2 ∂2E ∂t2 = 1 0 −1 c2 ∂2P ∂t2 + ∇(∇ · P) , (17) E = 1 2Ee −ı2(ωt−kz) + c.c., P = Pe−ı2(ωt−kz)+ c.c..
This wave equation differs from the conventional one [18] by the extra term ∇(∇ · P) 6= 0 which is not equal to zero because of the longitudinal contribution in P. We would like to point out that when the pump field is a plane wave there is only a longitudinal SH field
E k z (see Sec.II A). In this case the wave equation (14) becomes ∂2(
0E + P)/∂t2 = 0,
which is very different from eq.(17) because ∇(∇ · E) − ∇2E = (∂2/∂z2− ∂2/∂z2)E ≡ 0.
Now we are at the stage to simplify the wave equation (17). In our case the wave
propagates primarily along the z-axis with a small divergence angle (Fig. 2)
θ0 ≈ 1 2kw0 = w0 zR ∼ λ w0 1. (18)
Here λ is the wavelength of the pump field. We assume also that the pulse duration τ is
much longer than the period of field oscillations 2π/ω. This makes it possible to neglect
∂2E/∂z2 and ∂2E/∂t2 in eq. (17) (see Ref. [24]) and to get the following paraxial equation
for the SHG field ∂ ∂z + 1 c ∂ ∂t− ı 4k∆⊥ E = ı 2kf , kw0 1, τ ω 1, (19)
where ∆⊥ = ∇2ρ= ∂2/∂x2+ ∂2/∂y2 is the Laplacian operator over the transverse cartesian
longitudinal (fz) and transverse components (fρ) ˜ f = ˆz ˜fz+ ˆρ ˜fρ, (20) ˜ f = 1 0 −1 c2 ∂2P˜ ∂t2 + ∇(∇ · ˜P) ! ≈ 1 0 (2k)2P + ∇(∇ · ˜˜ P) , ˜ P = Pe−ı2(ωt−kz), ˜f = f e−ı2(ωt−kz)
Taking into account eqs.(11), (13) and (18) one can get the following expression for the transverse and longitudinal components of f
fρ = − ı8kρ 0w2 P, (21) fz = ı2kP 0(z2+ zR2) h2kρ2zR(ızR+ z) z2+ z2 R − ı4zR− z i .
One should point out that the origin of the fρ is the term ∇(∇ · ˜P) = ˆρ∂ρ(∂zP ) + · · · . A˜
simple estimation shows that the transverse contribution dominates: |fρ/fz| ∼ kw0 1.
As one can see from the paraxial equation (19) the transverse and longitudinal components of f generate, respectively, the transverse and longitudinal components of the SH field E.
C. Spatial distribution of the transverse and longitudinal SH fields. Radial polar-ization
It is convenient to write the solution of the paraxial equation (19) in terms of the retarded Green’s function (see Appendix B)
E(z, ρ, t) = 1 2π Z G(z − z0, ρ − ρ0, t − t0)f (z0, ρ0, t0)dz0dρ0dt0, (22) ∂ ∂z + 1 c ∂ ∂t − ı 4k∆⊥ G(z − z0, ρ − ρ0, t − t0) = δ(z − z0)δ(ρ − ρ0)δ(t − t0)Θ(t − t0). G(z − z0, ρ − ρ0, t − t0) = −ıδ t0− t − z 0− z c ) Θ(t − t0) k π(z − z0)exp ık|ρ − ρ 0)2 z − z0 , which guarantees that no contribution at remotely early times, t, before the source ˜
f (z0, ρ0, t0) = f (z0, ρ0, t0) exp(−ı2(ωt0 − kz0)) has been activated. Taking into account that
ˆ
the plane (ˆρ, ˆy) orthogonal to the z-axis using eq.(B5) 2π Z 0 dϕ[ˆρ0fρ(z0, ρ0, t0) + ˆzfz(z0, ρ0, t0)] exp ık|ρ − ρ 0)2 z − z0 (23) = 2π exp ık(ρ 2+ ρ02) z − z0 h − ˆρıfρ(z0, ρ0, t0)J1 2kρρ0 z − z0 + ˆzfz(z0, ρ0, t0)J0 2kρρ0 z − z0 i ,
where Jn(x) is a Bessel function. One can obtain the remaining integral over ρ0 using eq.(B5)
and get the following expressions for transverse and longitudinal contributions
E(z, ρ, t) = ˆρEρ(z, ρ, t) + ˆzEz(z, ρ, t), (24) Ei(z, ρ, t) = Ep(0)g 2 t − z c Ji(z, ρ), i = (ρ, z), where Jρ(z, ρ) = −2ρs0 z Z −∞ dz0 e Φ w4(z0)α2(z0), (25) Jz(z, ρ) = ı4πw0s0 (kw0)3 z Z −∞ dz0 e Φ w4(z0)α(z0) 2(ızR+ z0) w2(z0)α(z0) z − z0− 2k 2ρ2 α(z0) − (ı4zR+ z0) , Φ = ıkρ 2 z − z0 − ı2ψ(z 0 ) − k 2ρ2 (z − z0)α(z0), α(z 0 ) = 2(z − z 0) w2(z0) − ık z − z0 R(z0) + 1 , s0 = 8π G Γ − ıνN zRr01re
Here re = e2/(4π0mc2) = 2.82 × 10−13cm is the classical electron radius and G = E
(0)
p d01/~
is the Rabi frequency. It is important to notice that there is no transverse field on the beam axis
Eρ(z, ρ = 0, t) = 0. (26)
Eq.(24) indicates that the transverse SH field ˆρEρ is oriented along the radius ρ
perpendic-ular to the beam axis (Fig. 3). This means that the transverse field has radial polarization (see also Sec. III).
D. Role of photoabsorption
In the equations above the photoabsorpion of X-rays is ignored. This approximation is valid for the pump beam whose frequency is far from any resonance. In contrast, the SHG
field is in strict resonance with the dipole allowed transition |0i → |1i (1s − 3p for Ne and 1s − 4p for Ar). Therefore, this absorption channel should be taken into account. With the solution (24) at hand we are almost prepared to include the photoabsorption in the SH field. As shown in Appendix C the photoabsorption of the SHG field modifies only the integrands
at the right-hand side of equations (25) for Jρ(z, ρ) and Jz(z, ρ). Namely, these integrands
should be multiplied by the factor
exp z
0− z
2`
, (27)
where ` = 1/N σabs is the photoabsorption length while σabs is the resonant photoabsorption
cross section. According to simulations the photoabsorption length should be larger or comparable with the Rayleigh range
` ∼ zR (28)
to make it possible for the SHG field to reach the optimal value.
III. RESULTS OF SIMULATIONS AND DISCUSSION
We applied the theory outlined above to two atomic systems, Ne and Ar, under the strict
two-photon resonance (2ω = ω10) with 1s → 3p transitions for Ne and 1s → 4p transitions
for Ar. In the simulations, the peak pump intensity used was Ip(0) = c0|E
(0)
p |2/2 = 1016
W/cm2 (Fig. 2), and the following parameters were adopted for Ne: ~ω
1s−3p = 867.4 eV,
σabs(1s − 3p) = 1.5 × 10−18 cm2[25], 2~Γ = 0.27 eV [26], and for Ar: ~ω1s−4p= 3203.42 eV,
σabs(1s − 4p) = 0.12 × 10−18 cm2[27], 2~Γ = 0.66 eV[28]. The concentration of the atoms and
the beam waist were equal to N = 1019 cm−3 and w
0 = 1.0µm, respectively. The Rayleigh
range was zR≈ 103µm and zR ≈ 4 × 103µm for Ne and Ar, respectively. The corresponding
values of the photoabsorption lengths ` ≈ 0.67 × 103µm and 8 × 103µm satisfy the condition
(28).
We solved the paraxial equation with homogeneous distribution of the concentration. The
SHG radiation is characterized by the intensity distributions of the transverse (Iρ(z, ρ, t))
and longitudinal (Iz(z, ρ, t)) components of the SH field (24)
Ii(z, ρ, t) =
1
2c0|Ei(z, ρ, t)|
and by the energy conversion efficiency βi = Wi(z) Wp , Wi(z) = 2π Z ∞ 0 dt Z 2π 0 dρIi(z, ρ, t). (30)
First we studied the SH field neglecting the photoabsorption. In Fig. 4 and Fig. 5 we
display the spatial distribution of the SH intensities Iρ(z, ρ, t) and Iz(z, ρ, t) for Ne and Ar,
respectively. One can see that the transverse and longitudinal SH fields show very different
radial structures with Iρ= 0 on the axis of the beam ρ = 0, and the transverse field ˆρEρ(24)
has an unusual radial polarization as shown in Fig. 3. The transverse SH field Iρ is much
stronger than the longitudinal one Iz, and the energy conversion efficiency of the transverse
SH field is about four to five orders of magnitude larger than that of the longitudinal SH field as shown in Fig. 6.
Because of the deeper ionization potential and smaller core-electron transition dipole moment, the conversion efficiencies of the X-ray SH fields from Ar atomic vapour are much smaller than those from Ne (Fig. 6). However, when the photoabsorption of the generated SH fields is considered, the final conversion efficiencies from Ar and Ne become comparable. Below we investigated the conditions for the experimental observation of the SHG process with X-rays in atomic Ne and Ar vapours by taking into account the resonant one-photon absorption of the SH field during propagation. The photoabsorption changes the spatial distribution of the SH field (Fig. 7) and reduces the energy conversion efficiency in one order of magnitude for Ne and in four times for Ar, as one can see from Fig. 6 and Fig.8. As it is expected the SH field is confined in the focal region in the range limited by the photoabsorption length ` (see Fig. 7). Due to this circumstance the energy conversion
efficiency becomes maximal at z = zmax = 0.7 mm for Ne and zmax = 0.5 cm for Ar. This
range defines the size of the gas cell which should be around zmax.
IV. SUMMARY
In this paper, we investigated the second harmonic generation in systems with inversion symmetry in the X-ray region. Our theory is applied to SHG in neon and argon pumped by a strong X-ray field tuned in resonance with the two-photon transition 1s → 3p in Ne and 1s → 4p in Ar. The non-dipole population of these core-excited states is followed by the emission of the SH field. We describe the SHG in atoms in terms of a density
matrix formalism and paraxial equation taking into account the resonant photoabsorption of the SH radiation. In contrast to the plane wave pump field, the Gaussian pump beam generates transverse SH photons with radial polarization. By taking into account the X-ray photoabsorption effect, the energy conversion efficiencies to the transverse SH fields are
expected to be orders of 10−11and 10−12in Ne (867.4 eV) and Ar (3203.4 eV) atomic vapours
for the pump 1016 W/cm2, respectively.
Acknowledgements. The research leading to these results has received funding from a public grant from the Laboratoire d’Excellence Physics Atoms Light Matter (LabEx PALM), the Laboratoire d’Excellence MiChEm, and the Swedish Research Council (VR). JL thanks the support by the National Science Foundation of China under grant Nos. 11574082 and 11974108, and the Fundamental Research Funds for the Central Universities under grant No. 2018MS050. FG and SP acknowledge support within the State contract of the Russian Federation Ministry of Education and Science for Siberian Federal University for scientific
research in 2017-2019 (Project 3.2662.2017). H˚A acknowledges the Knut and Alice
Wallen-berg foundation for financial support (Grant No. KAW-2013.0020).
Appendix A: The pulsed Gaussian beam (11) is eigen function of paraxial operator
To get pulsed Gaussian beam (11) with the carrier frequency ω we need to convolute fundamental Gaussian mode
E(ω, t) = 1 2E(ω)e −ı(ωt−kz) + c.c., (A1) E(ω) = ˆxEp(0) w0 w(z) exp − % 2 w2(z) exp ı k % 2 2R(z) − ψ(z) ,
with the spectral distribution g(ω0− ω) = exp(−(ω0− ω)2τ2/2)/τ√2π centered at the carrier
frequency ω Ep(ω, t) = ∞ Z −∞ g(ω0 − ω)E(ω0, t)dω0. (A2)
The mode E(ω) is the eigen function of the stationary paraxial or Helmholtz equation ∂ ∂z − ı 2k∆⊥ E(ω) = 0. (A3)
The substitution of the fundamental mode (A1) in the convolution (A2) results in an ex-pression Ep(ω, t) = ˆ x 2E (0) p ∞ Z −∞ dω0g(ω0− ω) × exp −ıω0 t − z c − ρ2 2cR(z) w0 w(z)exp − % 2 w2(z) e−ıψ(z)+ c.c. (A4)
We neglect the ω0-dependence of the Rayleigh range zR = k0w02/2 ≈ kw02/2 because the
variation of the frequency ∆ω = |ω0− ω| ∼ 1/τ in the Fourier transform (A4) is negligibly small in comparison with the carrier frequency of X-ray pulse: ∆ω/ω ∼ 1/τ ω 1. Thus
Ep(ω, t) ≈ ˆ x 2E (0) p exp −ıω t − z c − ρ2 2cR(z) w0 w(z)exp − % 2 w2(z) e−ıψ(z) ×g t − z c − ρ2 2cR(z) + c.c., g(t) = exp − t 2 2τ2 (A5)
Within paraxial approximation [24] (kw0 1) we can neglect ρ2/2cR(z) in the Gaussian
g (t − z/c − ρ2/2cR(z))) because z ∼ zR∼ kw02 ρ2 2R(z) ∼ w2 0 zR ∼ 1 k, ρ2 2cR(z)τ ∼ 1 τ ω 1. (A6) However, we should keep ρ2/2cR(z) in the oscillatory term exp[−ıω(t − z/c − ρ2/2cR(z))]
because ω ρ 2 2cR(z) = kρ2 2R(z) ≈ kw2 0 2zR ∼ 1. (A7)
Finally we get eq.(11) for Ep(ω, t).
Now we are in stage to show that Ep from eq.(11) satisfies the paraxial equation
∂ ∂z + 1 c ∂ ∂t − ı 2k∆⊥ Ep= 0. (A8)
Let us apply the operator = −∇2+ ∂2/c2∂2t to both sides of eq.(A2)
Ep(ω, t) = ∞
Z
−∞
g(ω0 − ω)E(ω0, t)dω0. (A9) Using the paraxial approximation, kw0 1, τ ω 1 and eq.(A3) we get eq.(A8)
∂ ∂z + 1 c ∂ ∂t − ı 2k∆⊥ Ep = eı(ωt−kz) (A10) × ∞ Z −∞ dω0g(ω0− ω)k 0 ke −ı(ω0t−k0z) ∂ ∂z − ı 2k0∆⊥ E(ω0) = 0. (A11)
The paraxial approximation is broken when kw0 <∼ 1, τ ω ∼ 1. In this case one should<
restore in eq. (19) the second derivatives over z and time: ∂/∂z → ∂/∂z − (ı/4k)∂2/∂z2,
∆⊥ → ∆⊥ − ∂2/c2∂t2. However, such conditions are difficult to reach in X-ray region.
For example, the condition τ ω ∼ 1 corresponds to few cycle X-ray pulse, where the pulse duration is comparable with the period of the field oscillations.
Appendix B: Green’s function for the time-dependent paraxial equation
Let us find the Green’s function of non-stationary paraxial equation ∂ ∂z + 1 c ∂ ∂t− ı 2K∆⊥ G(z, ρ, t) = Θ(t)δ(t)δ(z)δ(ρ), (B1)
where ∆⊥ = ∂2/∂x2 + ∂2/∂y2, ρ = (x, y), δ(ρ) = δ(x)δ(y) and Θ(t) is the step function
which is equal to zero when t < 0. Taking the Fourier transform of the Green’s function and of the Dirac δ-functions we get
G(z, ρ, t) = Θ(t) (2π)4 ∞ Z −∞ dµ ∞ Z −∞ dν ∞ Z −∞ dp ∞ Z −∞ dqGµ,ν,p,qeıµt+ıνz+ıpx+ıqy, Gµ,ν,p,q= − 2ı µ c + ν + p2+q2 2K . (B2)
Keeping in mind that t ≥ 0 and taking the integral along half circle in upper half plane
around the pole µ = −cν + p22K+q2
∞ Z −∞ eıµt µ c + ν + p2+q2 2K dµ = −ıcπ exp −ıct ν + p 2+ q2 2K , (B3)
we obtain the following expression for the Green’s function G(z, ρ, t) = −ıδ t − z c Θ(t) K 2πzexp ıKρ 2 2z (B4) which allows to find the SHG field (22) with help of the following integrals[29]
J0(a) = 1 2π Z 2π 0 eıa cos θdθ, 1 2π Z 2π 0
cos θe−ıa cos θdθ = ı d
daJ0(a) = −ıJ1(a), (B5) ∞ Z 0 e−a2ρ2ρn+1Jn(bρ)dρ = bn (2a2)n+1e −b2 4a2, Re(a2) > 0.
Appendix C: Photoabsorption of SHG field
The strongest absorption channel is the absorption of SH field which is in resonance with |0i → |1i transition. To take into account this photoabsorption we need to add −E/2` at the right-hand side of paraxial equation (19)
∂ ∂z + 1 c ∂ ∂t− ı 4k∆⊥ E = −1 2`E + ı 2kf , (C1)
where ` = 1/σN is the length of resonant absorption of the SHG field with the
photoabsorp-tion cross secphotoabsorp-tion σ. Using the substituphotoabsorp-tion E = ˜E exp(−z/2`) one can see that ˜E satisfies
paraxial equation (19) ∂ ∂z + 1 c ∂ ∂t− ı 4k∆⊥ ˜ E = ı 2kf e z/2` (C2)
with modified source term. This equation has the solution given by eq.(22) with f replaced by
f exp(z0/2`). Taking into account this we get immediately the solution of paraxial equation
with photoabsorption (C1) E(z, ρ, t) = ˜Ee−z/2` = e −z/2` 2π Z G(z − z0, ρ − ρ0, t − t0)f (z0, ρ0, t0)ez0/2`dz0dρ0dt0. (C3)
This means that to include the photoabsorption we should multiply by exp((z0− z)/2`) the
integrand at the right-hand side of equations (25) for Jρ(z, ρ) and Jz(z, ρ).
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m=-1
0
1
1 = n p
0 = 1s
!
r
10↑↑ z ↑↑
k
!
FIG. 1: The dipole moment r10 of the 1s → np transition in atom is parallel to k. The axis of
- 4
- 2
0
2
4
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
1 0 0 5 . 6 2 E 3 3 . 1 6 E 5 1 . 7 8 E 7 1 . 0 0 E 9 5 . 6 2 E 1 0 3 . 1 6 E 1 2 1 . 7 8 E 1 4 1 . 0 0 E 1 6
(
µ
m
)
N e
I
p( z ,
)
0= w
0/ z
R~ 1 0
- 3- 1 5
- 1 0
- 5
0
5
1 0
1 5
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
1 0 0 5 . 6 2 E 3 3 . 1 6 E 5 1 . 7 8 E 7 1 . 0 0 E 9 5 . 6 2 E 1 0 3 . 1 6 E 1 2 1 . 7 8 E 1 4 1 . 0 0 E 1 6
(
µ
m
)
z ( c m )
A r
0= w
0/ z
R~ 0 . 2 5
×
1 0
- 3FIG. 2: 2D map of the pump intensity at t = z/c for Ne and Ar atomic vapours. The legend shows the intensity in W/cm2.
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
(
µ
m
)
0 . 1 0 0 0 . 4 2 2 1 . 7 8 7 . 5 0 3 1 . 6 1 3 3 5 6 2 2 . 3 7 E 3 1 . 0 0 E 4FIG. 3: Radial distribution of Iρfor Ne atomic vapour at z = 0.02 m. Black arrows show
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
(
µ
m
)
0 . 1 0 0 0 . 7 9 4 6 . 3 0 5 0 . 1 3 9 7 3 . 1 6 E 3 2 . 5 1 E 4 1 . 9 9 E 5 1 . 5 8 E 6I
( z ,
)
N e
- 4
- 2
0
2
4
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
1 . 0 0 E - 6 1 . 0 8 E - 5 1 . 1 6 E - 4 0 . 0 0 1 2 5 0 . 0 1 3 5 0 . 1 4 6 1 . 5 7 1 7 . 0 1 8 3
(
µ
m
)
z ( c m )
I
z( z ,
)
FIG. 4: 2D map of the SHG intensity at t = z/c of Ne atomic vapour. The photoabsorption is negleted. The legend shows the intensity in W/cm2.
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
1 . 0 0 E - 3 0 . 0 1 0 6 0 . 1 1 2 1 . 1 9 1 2 . 6 1 3 3 1 . 4 1 E 3 1 . 4 9 E 4 1 . 5 8 E 5
(
µ
m
)
A r
I
( z ,
)
- 1 5
- 1 0
- 5
0
5
1 0
1 5
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
1 . 0 0 E - 9 1 . 4 4 E - 8 2 . 0 7 E - 7 2 . 9 7 E - 6 4 . 2 8 E - 5 6 . 1 5 E - 4 0 . 0 0 8 8 5 0 . 1 2 7 1 . 8 3
(
µ
m
)
I
z( z ,
)
z ( c m )
FIG. 5: 2D map of the SHG intensity at t = z/c of Ar atomic vapour. The photoabsorption is neglected. The legend shows the intensity in W/cm2.
0 1 2 3 4 - 2 - 1 0 1 2 3 4 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 1 2 3 4 5 - 5 0 5 1 0 1 5 0 . 0 0 . 3 0 . 6 0 . 9 1 . 2 1 . 5 10 10 (z ) N e z ( c m ) 10 14 z (z ) A r 10 12 (z ) 10 17 z (z ) z ( c m )
FIG. 6: Energy conversion efficiencies of the transverse SH fields βρ(z) and the longitudinal SH
fields βz(z) in Ne and Ar atomic vapours. The photoabsorption is neglected. The vertical axises
show the energy conversion efficiencies multiplied by the factor 10n: For example, 1010βρ and
- 1 5 - 1 0 - 5 0 5 1 0 1 5 0 5 1 0 1 5 2 0
A r
0 . 0 1 0 0 0 . 0 3 9 2 0 . 1 5 4 0 . 6 0 2 2 . 3 6 9 . 2 5 3 6 . 3 1 4 2 5 5 7 2 . 1 8 E 3 8 . 5 6 E 3
(
µ
m
)
z ( c m )
- 4 - 2 0 2 4 0 5 1 0 1 5 2 0 0 . 0 1 0 0 0 . 0 5 5 5 0 . 3 0 8 1 . 7 1 9 . 4 7 5 2 . 5 2 9 1 1 . 6 2 E 3 8 . 9 7 E 3 4 . 9 8 E 4 2 . 7 6 E 5
(
µ
m
)
z ( c m )
N e
FIG. 7: Distribution of the transverse SH field Iρ for Ne and Ar by taking into account the
photoabsorption. The legend shows the intensity in W/cm2.
- 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 - 4 - 2 0 2 4 6 8 1 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 N e z ( c m ) 10 11 (z ) A r 10 12 (z ) z ( c m )
FIG. 8: Energy conversion efficiency βρ(z) for Ne and Ar by taking into account the
photoabsorp-tion. The red dashed lines display βρ(z) calculated by neglecting the photoabsorption (see Fig. 6).
