OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 12 — Jun. 9, 2008
  • pp: 8324–8331
« Show journal navigation

Nonlinear optics in the X-ray regime: nonlinear waves and self-action effects

C. Conti, A. Fratalocchi, G. Ruocco, and F. Sette  »View Author Affiliations


Optics Express, Vol. 16, Issue 12, pp. 8324-8331 (2008)
http://dx.doi.org/10.1364/OE.16.008324


View Full Text Article

Acrobat PDF (295 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We study the nonlinear refraction of X-rays in highly ionized condensed matter by using a classical model of a cold electron plasma in a lattice of still ions coupled with Maxwell equations. We discuss the existence and stability of nonlinear waves. As a real-world example, we consider beam self-defocusing in crystalline materials (B, C, Li, Na). We predict that nonlinear processes become comparable to the linear ones for focused beams with powers of the order of mc3/ro (≈10 GW), the classical electron power. As a consequence, nonlinear phenomena are expected in currently exploited X-ray Free-Electron Lasers and in their future developments.

© 2008 Optical Society of America

1. Introduction

2. Model and theoretical approach

X-ray propagation is described by Maxwell equations coupled to the hydrodynamic equations for a plasma of free electrons [12

12. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813–822 (1968). [CrossRef]

, 1

1. R. Dendy, Plasma physics: an in introductory course (Cambridge University Press, Cambridge, 1996).

]:

×h=j+ε0te,
×e=μ0th,
tn+·(nv)=0,
(t+v·)v+qm(e+μ0v×h)=0,
(1)

being e, h the time-dependent electromagnetic fields, v the electron mean velocity and j=-qn v the current generated by the charge density q(n-n 0), with n and n 0 the particle density of electrons and ions, respectively. We make the assumption that high energy photons instantaneously (i.e. at the attosecond scale) ionize the material, so that the electromagnetic field propagates in a periodically distributed cold plasma [14

14. A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, “Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals,” Phys. Rev. A 67, 041804-(4) (2003). [CrossRef]

, 4

4. H. Hora, Physics of Laser Driven Plasmas (Wiley-Interscience, New York, 1981).

]. The ion cores with density n 0, in other terms, are assumed to be “frozen”; they only form the crystal lattice. In our picture, the XFEL electric field amplitude is sufficiently high to heavily perturb the nuclear electric field and induce multiple ionization. We reduce (1) to an integrable ordinary differential equation (we defer extensive mathematical details to future publications); our procedure begins by writing the current as j=j L+j Δ, being j L a linear contribution and j Δ an arbitrary nonlinear term. In the linear regime j=j L and the Fourier domain (∂t→-), Eqs. (1) are written in the canonical form:

×H=iωε0(1ωα2ω2)E,×E=iωμ0H,
(2)

with ω 2 α=q 2 n 0/ 0. As detailed below, under typical experimental conditions (λ≈0.1 nm and n 0 ≈1030m-3), the term 1-δ, with δ=ω 2 α/ω 2, behaves as an effective periodic dielectric constant εr(r), whose period is settled by the crystal lattice constant a. The spectrum of such a periodic system is then found by applying the Floquet-Bloch theorem [19

19. K. Sakoda, Optical Properties of Photonic Crystals (Springer & Verlag, Berlin, 2001).

]. The eigenmodes of (2) are Bloch modes k, k satisfying the Hermitian eigenvalue problem:

HHk=(ωc)2Hk
(3)

being H=×1ε0εr× and ω(k) the medium dispersion relation. Actually, only magnetic modes are to be calculated since k depends on k via iωε 0 εr k=∇×k. Owing to the smallness of the dielectric perturbation ω 2 α/ω 2, the spectrum 𝓢={k, k, ω(k)} of the canonical structure is completely determined the representation theory of space groups [19

19. K. Sakoda, Optical Properties of Photonic Crystals (Springer & Verlag, Berlin, 2001).

, 20

20. T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990). [CrossRef]

]. Figure 1(a) shows the first Brillouin zone for the Face Centered Cubic (FCC) lattice, where the Brillouin zone is a truncated octahedron of height 4π/a. The dispersion relation is constructed from the free-space relation ω(k)=c|k|, folded over itself according to the dispersion periodicity ω(k)=ω(k+G) [20

20. T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990). [CrossRef]

], being G an arbitrary integer translation of the reciprocal lattice unit vectors (Fig. 1). No gap opens in the wave spectrum and Bloch modes exist for all frequencies ω. Each k point defines a set of degenerate plane waves, which possess the same eigenfrequency ω(k) in the absence of periodicity. If properly combined, these waves form a basis for Bloch modes, whose expression is provided by the theory of projection operators [20

20. T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990). [CrossRef]

]. Table 1 displays z-propagating magnetic Bloch modes for ωa/2πc<1 (e.g., for λ>a) for the FCC and the simple cubic (SC) lattice. Since Bloch modes are constructed from eigenwaves of the free-space, they are always doubly degenerate (i.e., they are either x or y polarized).

Once the spectrum 𝓢 of the canonical structure is known, the reduction of (1) is performed by exploiting the following identity (we let e=2ℜ[Eexp(-iωt)] and h=2ℜ[Hexp(-iωt)]:

·(E×H*+E*×H)+jΔ·E*=0.
(4)

Table 1. Expression of Bloch modes for SC and FCC lattices.

table-icon
View This Table

Equation (4) is a form of the reciprocity theorem of Maxwell equations [21

21. C. A. Balanis, Advanced Engineering Electromagnetics (Jhon Wiley & Sons, Toronto, 1989).

] and does not contain any approximation, as it can be verified by direct substitution. In the absence of nonlinear effects (j Δ=0) Eq. (4) is identically zero due to the linear relation between j L and the electric field; conversely when j Δ≠0 Eq. (4) allows to derive an exact evolution equation for the field. This requires the relationship between the current shift j Δ and E and H. To this aim we solve the plasma fluid equations by an iterative expansion in the field components: the overall current is written as j=j (1)+j (3), where j (1)=j L=(iq 2 n 0/ωm)E and j (3)=j Δ accounts for the third order nonlinearity. The second order terms can be neglected as far as the input direction does not satisfies phase matching conditions[14

14. A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, “Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals,” Phys. Rev. A 67, 041804-(4) (2003). [CrossRef]

]; correspondingly the phase-effect in the forward scattering (detailed below) is negligible. By letting

n=n0+2[n(1)eiωt]+2[n(2)e2iωt],
v=2[V(1)eiωt]+2[V(2)e2iωt],
(5)

we have (E 2E·E):

V(1)=iqωmE,n(1)=qω2mn0·E
(6)
V(2)=iq24ω3m2E2
(7)
n(2)=q22ω4m2[.(E2n0)+14.(n0E2)].
(8)

The nonlinear current is given by j (3)=-qn (2)[V (1)]*-q[n (1)]*V (2). In the final expression of j there are terms including ∇n 0, which account for out-of-axis nonlinear Bragg scattering.

3. Nonlinear waves and the classical electron power

In the following we will focus on the direct beam by retaining only terms that induce self-phase modulation in the input propagation direction. It is worthwhile remarking that the validity of our results is not limited to periodical samples: the features of the crystal lattice mainly influence diffracted orders, while the nonlinear effects on the forward propagating beam are expected to be observed for samples with comparable spatially-averaged charge density 〈n 0〉, i.e., independently on the specific lattice structure. Correspondingly, we only retain 〈n 0〉 terms in Eq. (6) and obtain

j(3)=iq4n08ω5m3E*2E2.
(9)

We consider a z-propagating beam with frequency below the first turning point of the band structure (see Fig. 1 for a/λ<1); in this case the treatment is simplified by the fact that only one Bloch function is involved. By exploiting Table 1, the field can be written as:

E=2η0A(z)ŷeikz,
H=2η0A(z)x̂eikz,
(10)
Fig. 1. (Color online) X-ray electromagnetic band-structure for a FCC lattice (the inset shows the first Brillouin zone).

where η0=μ0ε0 is the vacuum impedance. Bloch functions are normalized such that |A(z)|2 yields the beam intensity; correspondingly, Eq. (4) becomes

dAdz=iq4η02n08ω5m3A*e2ikzd2dz2(A2e2ikz).
(11)

We remark that Eq. (11) differs from the standard SPM equation (see e.g. [16

16. N. Bloembergen, “Wave propagation in nonlinear electromagnetic media,” Proceedings of the IEEE 51, 124–131 (1963). [CrossRef]

, 17

17. R. W. Boyd, Nonlinear Optics (Academic Press, New York, 2002).

, 22

22. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

]) and holds true for rapidly varying amplitudes A(z). Its solution is found as A(z)=A 0exp(iknlz), where knl is the nonlinear contribution to the wave-vector and satisfies

knl+q4η02A022ω5m3(k+knl)2=0.
(12)

Eq. (12) has always two real-valued solutions, as shown in Fig. 2(a) with γη 2 0 q 4n 0〉/8m 3 ω 5. At variance with standard SPM [17

17. R. W. Boyd, Nonlinear Optics (Academic Press, New York, 2002).

, 22

22. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

], the largest solution [continuous line in Fig. 2(a)] does not linearly grow with the power A 2 0 and asymptotically tends to -k from above. This wave is forward propagating. Conversely, the other solution provides k+knl<0 [dot-dashed line in Fig.2(a)], representing a backward propagating wave. These two solutions at large powers have the same wave-vector; this can involve a nonlinear coupling between them, which strongly relies on the material lattice and will be considered in future work.

The stability of the two nonlinear waves is found by letting A(z)=[A 0+a 1(z)]exp(iknlz), with a 1(z) a complex valued function. At the lowest order in a 1(z) we obtain

2γA02d2a1dz2+i[1+8γA02(k+knl)]da1dz+
[knl+8A02(k+knl)2γ]a1+4γ(k+knl)2a1*=0.
(13)

While looking for exponentially amplified disturbances a 1∝exp(αz), straightforward algebra leads to the conclusion that α is zero or imaginary for any value of A 0. The found solutions are therefore always stable. The existence of stable nonlinear plane waves allows to identify an effective refractive index n, which depends on the beam intensity. The latter can be written as n=1-δ++knl/k, being β the absorption coefficient. The term knl/k conversely accounts for the nonlinear wave-vector; at the lowest order in the intensity for the forward propagating solution it reads knl/k=-|n 2|I=-4γkI, with n 2=-4<0 the effective Kerr coefficient. To determine the values of I for which nonlinear effects become relevant, a noticeable figure of merit is the ratio between n 2 I and δ that, after simple algebra, is written as

Fig. 2. (Color online). (a) Solution of the nonlinear dispersion relation for the SPM Eq.(11), the continuous (dot-dashed) line is the forward (backward) propagating wave, the horizontal dashed line corresponds to the asymptotic value knl=-k; (b) divergence Vs input peak intensity I 0 in adimensional units.
Fig. 3. (Color online). (a) Atomic- and (b) electronic-loss-limited intensity for nonlinear defocusing.
𝒩n2Iδ=λ2I2πP0,
(14)

4. Self-defocusing and losses

We estimate the magnitude of nonlinear effects in the direct beam with reference to the simple case of self-defocusing. We consider a linearly polarized forward z-propagating beam with spatial profile E∝exp(-r 2 /w 2 0) focused on a sample (here r 2 =x 2+y 2 and w 0 is the beam waist). The field is expected to display a power-dependent divergence in the far field. For an input intensity A 2 0(r )=I 0 exp(-2r 2 /w 2 0), the output beam acquires an additional transverse nonlinear phase profile given by Φnl(r )=β[A 0(r )]L, being L the sample length. To the lowest order in the beam peak intensity I 0, the forward propagating solution of Eq. (12) gives β≅-4k 2 γA 2 0(r ). Since the most pronounced nonlinear effects are in proximity of the beam axis, one has

Φnl(r)4k2γLI0(12r2w02)
(15)

which implies that the sample acts as an effective thin lens with negative focal length

f=w0216kγI0L.
(16)

Correspondingly, the acquired divergence angle θ is given by

tan(θ)=w0f=16kγI0Lw0.
(17)

At any order in I 0, we have

f=2w02ξL(1+8ξ1+16ξ),
(18)

being ξkγI 0. Therefore, the divergence tan(θ) can be expressed as:

tan(θ)=L(1+8ξ1+16ξ)2w0ξ.
(19)

For large ξ, tan(θ) saturates at about 0.6L/w 0, which defines the largest attainable divergence [Fig. 2(b)]. In the future generation of X-ray FEL, peak intensities of the order of 100 GW with source size of 60 µm and divergence of 3 µrad will be at disposal [3

3. Techical Design Report of the European XFEL, DESY (2006-097). http://xfel.desy.de/tdr/tdr.

]. A beam with wavelength in the range 0.1-10 nm and intensity of ≈1022 W cm-2 (spot sizes ≈100 nm), displays a linear divergence of the order of 2 mrad; in order to have an observable nonlinear defocusing, we determine the intensity needed to induce a comparable beam divergence. For small divergence angles, owing to the linear relationship θw 0=16kγI 0 L between I 0 and the sample length L, the intensity needed to acquire a nonlinear divergence greater than the linear one is independent on the beam waist; this figure is therefore a good indicator to quantify nonlinear effects. In addition, in real materials the available effective length for the self-defocusing is compelled by photoelectric absorption and/or other material losses, whose strength strongly depends on the atomic number Z. In order to account for these effects, we consider the intensity needed to obtain a divergence comparable to the linear one on the sample characteristic loss length, the latter determined by two leading mechanisms: absorption [Fig. 3(a)] and electron-ion collisions [Fig. 3(b)] (electron-electron collisions are known to be negligible) [4

4. H. Hora, Physics of Laser Driven Plasmas (Wiley-Interscience, New York, 1981).

]. Figure 3(a) displays the intensity needed to induce a nonlinear divergence of 2 mrad for waist w 0=100 nm (or equivalently 20 mrad and for w 0=10 nm) by including the absorption as quantified by means of tabulated data [23

23. D. Vaughan, X-Ray Data Booklet (Lawrence Berkely Laboratory, Berkely, 1986). [CrossRef]

]. We estimated the average density of electrons as 〈n 0〉=Z𝒩aDM -1 m, where 𝒩a is the Avogadro number, D is the material density and Mm the molar mass, thus obtaining 〈n 0〉≅1030 m-3 for all of the considered materials. Figure 3(b) deals with electron-ion collisions described by the conductivity σcoll=3(KT)322Zm3πq2 , where K is the Boltzmann constant and T the plasma temperature. By comparing the two figures, one finds that electronic losses are dominant for wavelengths below 1 nm.

5. Conclusion

In conclusion, we predict that X-rays self-phase modulation is observable within the reported expected performances of future X-ray FEL sources for wavelengths of 1–10 nm, and in the current FLASH experiments at DESY. If tightly focused beams are used for retrieving structural information (e.g. in the foreseen single molecule diffraction experiments), we find that as far as the involved powers are of the order of P 0=mc 3/r 0≅10 GW, distortion due to nonlinear refraction could be observable and influence the interpretation of the scattering pattern. This result is expected to be independent on the specific structure of the molecule.

References and links

1.

R. Dendy, Plasma physics: an in introductory course (Cambridge University Press, Cambridge, 1996).

2.

T. Pfeifer, C. Spielmann, and G. Gerber, “Femtosecond x-ray science,” Rep. Prog. Phys. 69, 443–505 (2006). [CrossRef]

3.

Techical Design Report of the European XFEL, DESY (2006-097). http://xfel.desy.de/tdr/tdr.

4.

H. Hora, Physics of Laser Driven Plasmas (Wiley-Interscience, New York, 1981).

5.

R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, “Pair Creation and an X-Ray Free Electron Laser,” Phys. Rev. Lett. 87, 193902-(4) (2001). [CrossRef] [PubMed]

6.

R. Moshammer, Y. H. Jiang, L. Foucar, A. Rudenko, Th. Ergler, C. D. Schrter, S. Ldemann, K. Zrost, D. Fischer, J. Titze, T. Jahnke, M. Schffler, T. Weber, R. Drner, T. J. M. Zouros, A. Dorn, T. Ferger, K. U. Khnel, S. Dsterer, R. Treusch, P. Radcliffe, E. Plnjes, and J. Ullrich, “Few-Photon Multiple Ionization of Ne and Ar by Strong Free-Electron-Laser Pulses,” Phys. Rev. Lett. 98, 203001-(4) (2007). [CrossRef] [PubMed]

7.

H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, “Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation,” Phys. Rev. Lett. 94, 023001-(4) (2005). [CrossRef] [PubMed]

8.

Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, “Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy,” Phys. Rev. Lett. 94, 043001-(4) (2005). [CrossRef] [PubMed]

9.

T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, “Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes,” Phys. Rev. A 72, 023409-(8) (2005). [CrossRef]

10.

E. P. Benis, D. Charalambidis, T. N. Kitsopoulos, G. D. Tsakiris, and P. Tzallas, “Two-photon double ionization of rare gases by a superposition of harmonics,” Phys. Rev. A 74, 051402(R)-(4) (2006). [CrossRef]

11.

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature (London) 432, 605–608 (2004). [CrossRef] [PubMed]

12.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813–822 (1968). [CrossRef]

13.

S. S. Jha and C. S. Warke, “Nonlinear electromagnetic response of Bloch electrons in a magnetic field,” Nuovo Cimento 53B, 120–135 (1968).

14.

A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, “Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals,” Phys. Rev. A 67, 041804-(4) (2003). [CrossRef]

15.

S. S. Jha and J. W. F. Woo, “Nonlinear Response of Bound Electrons to X Rays,” Phys. Rev. B 5, 4210–4212 (1972). [CrossRef]

16.

N. Bloembergen, “Wave propagation in nonlinear electromagnetic media,” Proceedings of the IEEE 51, 124–131 (1963). [CrossRef]

17.

R. W. Boyd, Nonlinear Optics (Academic Press, New York, 2002).

18.

R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond X-ray pulses,” Nature (London) 406, 752–757 (2000). [CrossRef] [PubMed]

19.

K. Sakoda, Optical Properties of Photonic Crystals (Springer & Verlag, Berlin, 2001).

20.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990). [CrossRef]

21.

C. A. Balanis, Advanced Engineering Electromagnetics (Jhon Wiley & Sons, Toronto, 1989).

22.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

23.

D. Vaughan, X-Ray Data Booklet (Lawrence Berkely Laboratory, Berkely, 1986). [CrossRef]

OCIS Codes
(160.4330) Materials : Nonlinear optical materials
(190.5940) Nonlinear optics : Self-action effects
(340.0340) X-ray optics : X-ray optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: January 18, 2008
Revised Manuscript: March 11, 2008
Manuscript Accepted: March 18, 2008
Published: May 23, 2008

Citation
C. Conti, A. Fratalocchi, G. Ruocco, and F. Sette, "Nonlinear optics in the X-ray regime: nonlinear waves and self-action effects," Opt. Express 16, 8324-8331 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-8324


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. R. Dendy, Plasma physics: an in introductory course (Cambridge University Press, Cambridge, 1996).
  2. T. Pfeifer, C. Spielmann, and G. Gerber, "Femtosecond x-ray science," Rep. Prog. Phys. 69,443-505 (2006). [CrossRef]
  3. Techical Design Report of the European XFEL, DESY (2006-097). http://xfel.desy.de/tdr/tdr.
  4. H. Hora, Physics of Laser Driven Plasmas (Wiley-Interscience, New York, 1981).
  5. R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, "Pair Creation and an X-Ray Free Electron Laser, " Phys. Rev. Lett. 87, 193902-(4) (2001). [CrossRef] [PubMed]
  6. R. Moshammer Y. H. Jiang, L. Foucar, A. Rudenko, Th. Ergler, C. D. Schrter, S. Ldemann, K. Zrost, D. Fischer, J. Titze, T. Jahnke, M. Schffler, T. Weber, R. Drner, T. J. M. Zouros, A. Dorn, T. Ferger, K. U. Khnel, S. Dsterer, R. Treusch, P. Radcliffe, E. Plnjes, and J. Ullrich, " Few-Photon Multiple Ionization of Ne and Ar by Strong Free-Electron-Laser Pulses," Phys. Rev. Lett. 98, 203001-(4) (2007). [CrossRef] [PubMed]
  7. H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005). [CrossRef] [PubMed]
  8. Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, " Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy," Phys. Rev. Lett. 94, 043001-(4) (2005). [CrossRef] [PubMed]
  9. T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005). [CrossRef]
  10. E. P. Benis, D. Charalambidis, T. N. Kitsopoulos, G. D. Tsakiris, and P. Tzallas, " Two-photon double ionization of rare gases by a superposition of harmonics," Phys. Rev. A 74, 051402(R)-(4) (2006). [CrossRef]
  11. T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, "Nonlinear optics in the extreme ultraviolet," Nature (London) 432,605-608 (2004). [CrossRef] [PubMed]
  12. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, "Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry, " Phys. Rev. 174,813-822 (1968). [CrossRef]
  13. S. S. Jha, and C. S. Warke, "Nonlinear electromagnetic response of Bloch electrons in a magnetic field," Nuovo Cimento 53B,120-135 (1968).
  14. A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, "Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals," Phys. Rev. A 67, 041804-(4) (2003). [CrossRef]
  15. S. S. Jha, and J. W. F. Woo, "Nonlinear Response of Bound Electrons to X Rays," Phys. Rev. B 5,4210-4212 (1972). [CrossRef]
  16. N. Bloembergen, "Wave propagation in nonlinear electromagnetic media," Proceedings of the IEEE 51,124-131 (1963). [CrossRef]
  17. R. W. Boyd, Nonlinear Optics (Academic Press, New York, 2002).
  18. R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature (London) 406,752-757 (2000). [CrossRef] [PubMed]
  19. K. Sakoda, Optical Properties of Photonic Crystals (Springer & Verlag, Berlin, 2001).
  20. T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990). [CrossRef]
  21. C. A. Balanis, Advanced Engineering Electromagnetics (Jhon Wiley & Sons, Toronto, 1989).
  22. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
  23. D. Vaughan, X-Ray Data Booklet (Lawrence Berkely Laboratory, Berkely, 1986). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 

Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited