- Email: [email protected]
Generation of single-cycle attosecond pulses in the vacuum ultraviolet
1 March 1998
Optics Communications 148 Ž1998. 75–78
Generation of single-cycle attosecond pulses in the vacuum ultraviolet Ivan P. Christov 1, Margaret M. Murnane, Henry C. Kapteyn
)
Center for Ultrafast Optical Science, UniÕersity of Michigan, Ann Arbor, MI 48109-2099, USA Received 25 August 1997; revised 23 October 1997; accepted 24 October 1997
Abstract A model for the interaction of a focused electron beam and a terawatt femtosecond optical pulse has been developed which demonstrates that single-cycle VUV pulses, with durations below 0.5 fs, can be generated. The ultrashort VUV pulses result from the nonlinear motion of the electrons in a ponderomotive force induced by a tightly focused femtosecond pump pulse. Our results demonstrate that the nonlinear properties of relativistic free electrons can be exploited to generate useful sources of light in the VUV. q 1998 Published by Elsevier Science B.V.
In the past few years, there has been rapid progress on the generation of ultrashort sub-20 fs duration pulses in the visible and X-ray regions w1–6x. Extending ultrashort pulse sources to even shorter durations Žsub-fs. is difficult, but several potential techniques have been proposed recently. These techniques have considered the possibility of synthesizing attosecond pulses by a Fourier superposition of spectral components of a properly phased spectrum, in a manner similar to that in modelocked lasers. One approach proposes to superpose cw lasers of different wavelengths with computer-controlled relative phase, in order to generate sub-femtosecond pulses w7x. However, the number of available frequencies is limited, so that the output would consist of a series of attosecond spikes at extremely high repetition rate. Other approaches use high-harmonic generation ŽHHG., which occurs in atoms under the influence of a strong laser field w8–11x. These approaches assume that the harmonics emitted have a well-defined relative phase, and when superimposed can give an attosecond pulse. However, due to locally varying fields and other effects
)
Corresponding author. E-mail: [email protected] Permanent address: Department of Physics, Sofia University, 1126 Sofia, Bulgaria. 1
w12x, this may not be the case. Other novel approaches to generating sub-femtosecond pulses use cascaded stimulated Raman scattering w13x, or bremsstrahlung radiation by ultrashort electron pulses w14x. The most important criterion for generating sub-fs pulses is a sufficiently high degree of spatio-temporal coherence in the source. In the simplest case, this implies that all particles which take part in the nonlinear process must experience the same pump field, and the emission must arise from a spatial region smaller than the duration of the expected pulse. In this paper, we consider the case where the highly-coherent ‘nonlinear medium’ is an electron beam, which is focused to sub-micrometer spot sizes. Clearly, free electrons are a very promising medium for high-intensity laser–matter interactions, since they are not influenced by the atomic potential, so that the electron beam appears as essentially a non-dispersive medium. Moreover, the electron beam can be tightly focused while still maintaining a very high density, which can greatly exceed the density of atoms in the gas nozzles used for HHG. When the free electrons interact with a strong optical wave, they accelerate to relativistic velocities, where their motion becomes a nonlinear function of the incident field. The electrons then radiate discrete harmonics of the pump frequency w15,16x. Particularly good results are expected when the electrons are driven by a strong femtosec-
0030-4018r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 6 4 1 - X
I.P. ChristoÕ et al.r Optics Communications 148 (1998) 75–78
76
Fig. 1. Set-up for the generation of VUV attosecond pulses: Ža. single focused pulse; Žb. a pair of p-shifted co-propagating focused pulses. P is the observation point.
ond pulse, since in this case, the scattered radiation consists of only a few ultrashort pulses w17x 2. In our scheme, an attosecond pulse is generated as a result of relativistic Thomson scattering by a focused electron beam, which co-propagates with a tightly focused high-intensity femtosecond laser pulse. First we consider a collinear geometry, where the optical and the electron beam co-propagate in the negative y direction, as shown in Fig. 1Ža.. The collinear geometry allows the creation of a homogenous optical field in the interaction region, when compared to the case of transverse propagation of the two beams. We calculate the electron motion by numerically solving the relativistic equation of motion of the electron w18x: dz
e s
dt
m
(
1y
Õ2 c
2
Ey
zŽ z P E . c
2
z=H q c
,
the negative y axis Žsee Fig. 1Ža... Clearly, the electron trajectory, r Ž t ., lies in Ž x, y . plane. First, we assume that the cross-section of the electron beam is much smaller than the size of the focused optical beam, as shown in Fig. 1Ža.. This should not be a problem for typical electron guns with an energy of ( 20 keV, where the diameter of the electron beam can be less than 10 nm w19x. During the interaction with the optical pulse Ž l s 800 nm., each electron experiences both the electric field Žwhich causes oscillations along the x-axis in Fig. 1., and the magnetic field Žwhich leads to a drift in yy direction. of the pump. Due to the spatial profile of the focused optical beam, the electrons also experience a ponderomotive force, proportional to the gradient of the intensity, which expels the electrons from the axis Žsee Fig. 2.. Since the ponderomotive force is near zero on the axis, the electrons close to the axis will remain in the beam and experience the entire optical pulse. Thus it would be difficult to obtain a very short scattered pulse. On the other hand, for electrons far from the axis of the optical beam, the ponderomotive force is small because the field is weak. Therefore there is an optimal offset between the electron beam and the axis of the focused optical beam where the ponderomotive force has a maximum, and expels all the electrons in a small electron beam out of the focus. Thus, if the laser pulse is sufficiently short, the electrons experience the electromagnetic field of the light only for a few optical cycles, and the scattered pulse can be extremely short. When an electron is under the influence of a very strong optical pulse, large changes in the scattered radiation occur w17x. Fig. 2 shows the changes in the Thomson scattering when a single electron is positioned at a distance x s y1.4 mm from the axis of a beam of diameter of 5 mm. The maximum pump pulse intensity is 5.9 = 10 19
Ž1.
where z is the velocity of the electron, E Ž r, t . and H Ž r, t . are the electric and magnetic components of the incident field Žsee Fig. 1., e and m are the charge and mass of the electron. In the far-field approximation, the radiated field detected by an observer at time t is w18x: Er Ž t . A
n = w w n y z Ž tX . rc x = z˙ Ž tX . x
w 1 y n P z Ž tX . rc x 3
.
Ž2.
Here n defines the observation direction, and all variables have to be evaluated at the retarded time tX Žgiven by the solution of the equation tX y r Ž tX .rc s t .. We assume that the optical pulse is linearly polarized and propagates along
2
The intensities in this reference have to be reduced by a factor of 100 in order to obtain the values in the laboratory frame.
Fig. 2. Angular dependence of the energy W Ž u . scattered by a single electron; dashed line – weak incident field; solid line – input intensity of 5.9=10 19 Wrcm2 . The duration of the input pulse is 7 fs, and the beam diameter in the focus is 5 mm. The electron beam is offset by y1.4 mm from the axis of the optical beam.
I.P. ChristoÕ et al.r Optics Communications 148 (1998) 75–78
Wrcm2 , and the duration is 7 fs FWHM. Almost all of the energy scattered by the electron is directed within a narrow cone, due to both the relativistic motion of the electron, and the effect of the ponderomotive force. Our simulations show that there is an optimal relation between the input intensity, the beam cross-section, and the electron beam position, where the direction of the radiated peak is close to the direction of electron motion Žsee Fig. 1Ža... Thus, for an observer looking towards the incoming electron, the scattered radiation appears Doppler upshifted. Although there is a time lag between the pulses scattered by different electrons in the beam, this can be easily compensated using a properly adjusted diffraction grating. In the geometry shown in Fig. 1Ža., the radiation emerging from different portions of the electron beam is affected by the divergence of the pump beam Žin y direction.. For example, for a Gaussian beam with a radius wo , the position where the ponderomotive force is maximum is x s wo w1 q Ž yrb . 2 x 0.5r2, where b is the confocal parameter of the beam w20x. Thus, different electrons experience different ponderomotive forces along the y-axis, and as a result, the radiation can vary significantly. Subsequent superposition of these radiated pulses would produce random spikes in time, instead of a single ultrashort pulse. Because of these limitations a second geometry was studied, where the pump field is formed by an equal pair of co-propagating pulses, with a p-phase shift between them. Due to destructive interference the region between the beams, a channel-like region is formed, which ensures a more homogenous distribution of the pump field along the y-axis ŽFig. 1Žb... Also, the spatial profile in the area of overlap has a higher gradient than that of a single beam, which further increases the ponderomotive force. Moreover, it can be shown that for the geometry shown in Fig. 1Žb., the ponderomotive force decreases as yy6 , while in the case of a single optical beam the dependence is yy4 . Therefore, the two beam geometry offers the possibility of stronger localization of the nonlinear interaction in both the transverse and longitudinal directions, significantly improving the coherence properties of the scattered radiation. The scattered pulse in the two-pump-beam scheme was simulated using a model electron beam of 100 electrons, which cover a distance of "40 mm in the y-direction, and move with a velocity of 0.3c Ženergy ( 20 keV.. The two pump pulses, with intensities of 5.9 = 10 19 Wrcm2 , and durations of 7 fs, at a wavelength of 800 nm, are each focused to a 5 mm spot-size. The two beams are offset from each other by 2 mm. The direction of observation is u s 4 rad Žsee Fig. 3.. For each individual electron, Eq. Ž1. gives the velocity zŽ t . as a function of the absolute time t. The acceleration is then calculated, which together with the velocity, is converted to the retarded time tX. For a given direction of observation, Eq. Ž2. predicts the detected field in the laboratory frame. Fig. 3Ža. shows the total electric field of the optical pulse scattered by the
77
electron beam. For the set of parameters used, the scattered pulse is single-cycle, with duration under 0.5 fs. Our simulations show that such short pulses can be generated by those electrons which are accelerated by the optical pulse to velocities above 0.98c. The time lag between the radiated pulses from electrons in different regions of the beam is compensated for using a diffraction grating with a ruling of 3400 linesrmm. For this grating, the time delay between two successive grooves is about 0.7 fs, which exceeds the duration of the pulses radiated by each separate electron. This ensures that the grating performs as a pure delay line Žnot as a spectrally dispersing device.. The corresponding spectrum is shown in Fig. 3Žb.. It is centered close to the 10th harmonic of the pump, i.e. 80 nm. The spectrum is continuous and should result in a single attosecond pulse, not an attosecond pulsetrain. Similar results were obtained using lower peak intensities, lower electron beam energies Ž- 1 keV., or when the position of the electron beam was 300 nm of its optimum position. Thus, the exact experimental conditions are flexible. From a practical viewpoint, recent work on ultrashortpulse Ti:sapphire amplifiers w21,22x has generate terawatt pulses with durations f 20 fs, focusable to intensities ) 10 18 Wrcm2. Even shorter pulses can be generated at lower energies w3x. By combining these technologies, TW pulses with durations under 10 fs can be expected in the near future. Also, to compare the acceleration of an electron in the process of HHG, and the relativistic case
Fig. 3. Ža. Total radiated field, and Žb. spectrum from an electron beam with 100 electrons.
78
I.P. ChristoÕ et al.r Optics Communications 148 (1998) 75–78
discussed herein, we note that due to the stronger field strength utilized here, the acceleration and radiated field is 1000 stronger than in the case of HHG, which is already a useful source of photons. Thus, even with a factor of 10 6 lower electron density, the pulse scattered by the electron beam would be comparable to the intensity of the pulse scattered by an atomic beam with density f 10 18 cmy3, typical of HHG experiments. Taking into account that the HHG intensity is many times weaker than the fundamental scattered light Žin contrast to the relativistic case., even with a low-current electron beam, the conversion efficiency in the relativistic case would be much higher. On the other hand, possible space charge and refractive index effects may limit the upper value of the current of the focused electron beam. Since the technique considered here uses electron beams with diameters up to 1 mm, the total number of electrons may significantly exceed that used in electron lithography. The value of the accessible current depends on the specific design of the electron gun and its focusing optics. In summary, our simulations show that the interaction of a focused electron beam with a femtosecond TW optical pulse can produce single-cycle attosecond pulses in the VUV. If high current density electron guns are used, the expected energy of the attosecond pulse generated by this technique can significantly exceed the energy of attosecond pulses produced by alternate schemes, including highharmonic generation in noble gases. Moreover, the expected scatter beam duration is single cycle in the VUV region of the spectrum.
Acknowledgements The authors gratefully acknowledge support from the National Science Foundation. H.K. acknowledges support from a Sloan Foundation Fellowship.
References w1x J. Zhou, G. Taft, C.P. Huang, M.M. Murnane, H.C. Kapteyn, I. Christov, Optics Lett. 19 Ž1994. 1149. w2x J. Zhou, J. Peatross, M. Murnane, H. Kapteyn, I. Christov, Phys. Rev. Lett 76 Ž1996. 752. w3x M. Nisoli, S.D. Silvestri, O. Svelto, Appl. Phys. Lett. 68 Ž1996. 2793. w4x I.P. Christov, V. Stoev, M. Murnane, H. Kapteyn, Optics Lett. 21 Ž1996. 1493. w5x I.P. Christov, J.P. Zhou, J. Peatross, A. Rundquist, M.M. Murnane, H.C. Kapteyn, Phys. Rev. Lett. 77 Ž1996. 1743. w6x S. Backus, J. Peatross, M. Murnane, H. Kapteyn, Optics Lett. 21 Ž1996. 665. w7x T.W. Hansch, Optics Comm. 80 Ž1990. 71. w8x G. Farkas, C. Toth, Phys. Lett. A 168 Ž1992. 447. w9x P.B. Corkum, N.H. Burnett, M.Y. Ivanov, Optics Lett. 19 Ž1994. 1870. w10x M. Protopapas, D.G. Lappas, C.H. Keitel, P.L. Knight, Phys. Rev. A 53 Ž1996. 2933. w11x P. Antoine, A. L’Huiller, M. Lewenstein, Phys. Rev. Lett. 77 Ž1996. 1234. w12x S.C. Rae, K. Burnett, J. Cooper, Phys. Rev. A 50 Ž1994. 3438. w13x A.E. Kaplan, Phys. Rev. Lett. 73 Ž1994. 1243. w14x S.D. Vartak, N.M. Lawandy, Optics Comm. 120 Ž1995. 184. w15x E.S. Sarachik, G.T. Schappert, Phys. Rev. D 1 Ž1970. 2738. w16x J.H. Eberly, E. Wolf, Progress in Optics, vol. 7, North Holland, Amsterdam, 1969. w17x I.P. Christov, M.B. Danailov, Optics Comm. 68 Ž1988. 291. w18x L. Landau, E. Lifshitz, The Classical Theory of Fields, 2nd ed, ch. 9, Addison-Wesley, Reading, MA, 1951. w19x J.R. Pierce, Theory and Design of Electron Beams, 2nd ed., ch. 6, Van Nostrand, Princeton, 1954. w20x A.E. Siegman, Lasers, University Science Books, Mill Valley, 1986. w21x J.P. Zhou, C.P. Huang, M.M. Murnane, H.C. Kapteyn, Optics Lett. 20 Ž1995. 64. w22x C.P. Barty et al., Optics Lett. 21 Ž1996. 668.
Optics Communications 148 Ž1998. 75–78
Generation of single-cycle attosecond pulses in the vacuum ultraviolet Ivan P. Christov 1, Margaret M. Murnane, Henry C. Kapteyn
)
Center for Ultrafast Optical Science, UniÕersity of Michigan, Ann Arbor, MI 48109-2099, USA Received 25 August 1997; revised 23 October 1997; accepted 24 October 1997
Abstract A model for the interaction of a focused electron beam and a terawatt femtosecond optical pulse has been developed which demonstrates that single-cycle VUV pulses, with durations below 0.5 fs, can be generated. The ultrashort VUV pulses result from the nonlinear motion of the electrons in a ponderomotive force induced by a tightly focused femtosecond pump pulse. Our results demonstrate that the nonlinear properties of relativistic free electrons can be exploited to generate useful sources of light in the VUV. q 1998 Published by Elsevier Science B.V.
In the past few years, there has been rapid progress on the generation of ultrashort sub-20 fs duration pulses in the visible and X-ray regions w1–6x. Extending ultrashort pulse sources to even shorter durations Žsub-fs. is difficult, but several potential techniques have been proposed recently. These techniques have considered the possibility of synthesizing attosecond pulses by a Fourier superposition of spectral components of a properly phased spectrum, in a manner similar to that in modelocked lasers. One approach proposes to superpose cw lasers of different wavelengths with computer-controlled relative phase, in order to generate sub-femtosecond pulses w7x. However, the number of available frequencies is limited, so that the output would consist of a series of attosecond spikes at extremely high repetition rate. Other approaches use high-harmonic generation ŽHHG., which occurs in atoms under the influence of a strong laser field w8–11x. These approaches assume that the harmonics emitted have a well-defined relative phase, and when superimposed can give an attosecond pulse. However, due to locally varying fields and other effects
)
Corresponding author. E-mail: [email protected] Permanent address: Department of Physics, Sofia University, 1126 Sofia, Bulgaria. 1
w12x, this may not be the case. Other novel approaches to generating sub-femtosecond pulses use cascaded stimulated Raman scattering w13x, or bremsstrahlung radiation by ultrashort electron pulses w14x. The most important criterion for generating sub-fs pulses is a sufficiently high degree of spatio-temporal coherence in the source. In the simplest case, this implies that all particles which take part in the nonlinear process must experience the same pump field, and the emission must arise from a spatial region smaller than the duration of the expected pulse. In this paper, we consider the case where the highly-coherent ‘nonlinear medium’ is an electron beam, which is focused to sub-micrometer spot sizes. Clearly, free electrons are a very promising medium for high-intensity laser–matter interactions, since they are not influenced by the atomic potential, so that the electron beam appears as essentially a non-dispersive medium. Moreover, the electron beam can be tightly focused while still maintaining a very high density, which can greatly exceed the density of atoms in the gas nozzles used for HHG. When the free electrons interact with a strong optical wave, they accelerate to relativistic velocities, where their motion becomes a nonlinear function of the incident field. The electrons then radiate discrete harmonics of the pump frequency w15,16x. Particularly good results are expected when the electrons are driven by a strong femtosec-
0030-4018r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 6 4 1 - X
I.P. ChristoÕ et al.r Optics Communications 148 (1998) 75–78
76
Fig. 1. Set-up for the generation of VUV attosecond pulses: Ža. single focused pulse; Žb. a pair of p-shifted co-propagating focused pulses. P is the observation point.
ond pulse, since in this case, the scattered radiation consists of only a few ultrashort pulses w17x 2. In our scheme, an attosecond pulse is generated as a result of relativistic Thomson scattering by a focused electron beam, which co-propagates with a tightly focused high-intensity femtosecond laser pulse. First we consider a collinear geometry, where the optical and the electron beam co-propagate in the negative y direction, as shown in Fig. 1Ža.. The collinear geometry allows the creation of a homogenous optical field in the interaction region, when compared to the case of transverse propagation of the two beams. We calculate the electron motion by numerically solving the relativistic equation of motion of the electron w18x: dz
e s
dt
m
(
1y
Õ2 c
2
Ey
zŽ z P E . c
2
z=H q c
,
the negative y axis Žsee Fig. 1Ža... Clearly, the electron trajectory, r Ž t ., lies in Ž x, y . plane. First, we assume that the cross-section of the electron beam is much smaller than the size of the focused optical beam, as shown in Fig. 1Ža.. This should not be a problem for typical electron guns with an energy of ( 20 keV, where the diameter of the electron beam can be less than 10 nm w19x. During the interaction with the optical pulse Ž l s 800 nm., each electron experiences both the electric field Žwhich causes oscillations along the x-axis in Fig. 1., and the magnetic field Žwhich leads to a drift in yy direction. of the pump. Due to the spatial profile of the focused optical beam, the electrons also experience a ponderomotive force, proportional to the gradient of the intensity, which expels the electrons from the axis Žsee Fig. 2.. Since the ponderomotive force is near zero on the axis, the electrons close to the axis will remain in the beam and experience the entire optical pulse. Thus it would be difficult to obtain a very short scattered pulse. On the other hand, for electrons far from the axis of the optical beam, the ponderomotive force is small because the field is weak. Therefore there is an optimal offset between the electron beam and the axis of the focused optical beam where the ponderomotive force has a maximum, and expels all the electrons in a small electron beam out of the focus. Thus, if the laser pulse is sufficiently short, the electrons experience the electromagnetic field of the light only for a few optical cycles, and the scattered pulse can be extremely short. When an electron is under the influence of a very strong optical pulse, large changes in the scattered radiation occur w17x. Fig. 2 shows the changes in the Thomson scattering when a single electron is positioned at a distance x s y1.4 mm from the axis of a beam of diameter of 5 mm. The maximum pump pulse intensity is 5.9 = 10 19
Ž1.
where z is the velocity of the electron, E Ž r, t . and H Ž r, t . are the electric and magnetic components of the incident field Žsee Fig. 1., e and m are the charge and mass of the electron. In the far-field approximation, the radiated field detected by an observer at time t is w18x: Er Ž t . A
n = w w n y z Ž tX . rc x = z˙ Ž tX . x
w 1 y n P z Ž tX . rc x 3
.
Ž2.
Here n defines the observation direction, and all variables have to be evaluated at the retarded time tX Žgiven by the solution of the equation tX y r Ž tX .rc s t .. We assume that the optical pulse is linearly polarized and propagates along
2
The intensities in this reference have to be reduced by a factor of 100 in order to obtain the values in the laboratory frame.
Fig. 2. Angular dependence of the energy W Ž u . scattered by a single electron; dashed line – weak incident field; solid line – input intensity of 5.9=10 19 Wrcm2 . The duration of the input pulse is 7 fs, and the beam diameter in the focus is 5 mm. The electron beam is offset by y1.4 mm from the axis of the optical beam.
I.P. ChristoÕ et al.r Optics Communications 148 (1998) 75–78
Wrcm2 , and the duration is 7 fs FWHM. Almost all of the energy scattered by the electron is directed within a narrow cone, due to both the relativistic motion of the electron, and the effect of the ponderomotive force. Our simulations show that there is an optimal relation between the input intensity, the beam cross-section, and the electron beam position, where the direction of the radiated peak is close to the direction of electron motion Žsee Fig. 1Ža... Thus, for an observer looking towards the incoming electron, the scattered radiation appears Doppler upshifted. Although there is a time lag between the pulses scattered by different electrons in the beam, this can be easily compensated using a properly adjusted diffraction grating. In the geometry shown in Fig. 1Ža., the radiation emerging from different portions of the electron beam is affected by the divergence of the pump beam Žin y direction.. For example, for a Gaussian beam with a radius wo , the position where the ponderomotive force is maximum is x s wo w1 q Ž yrb . 2 x 0.5r2, where b is the confocal parameter of the beam w20x. Thus, different electrons experience different ponderomotive forces along the y-axis, and as a result, the radiation can vary significantly. Subsequent superposition of these radiated pulses would produce random spikes in time, instead of a single ultrashort pulse. Because of these limitations a second geometry was studied, where the pump field is formed by an equal pair of co-propagating pulses, with a p-phase shift between them. Due to destructive interference the region between the beams, a channel-like region is formed, which ensures a more homogenous distribution of the pump field along the y-axis ŽFig. 1Žb... Also, the spatial profile in the area of overlap has a higher gradient than that of a single beam, which further increases the ponderomotive force. Moreover, it can be shown that for the geometry shown in Fig. 1Žb., the ponderomotive force decreases as yy6 , while in the case of a single optical beam the dependence is yy4 . Therefore, the two beam geometry offers the possibility of stronger localization of the nonlinear interaction in both the transverse and longitudinal directions, significantly improving the coherence properties of the scattered radiation. The scattered pulse in the two-pump-beam scheme was simulated using a model electron beam of 100 electrons, which cover a distance of "40 mm in the y-direction, and move with a velocity of 0.3c Ženergy ( 20 keV.. The two pump pulses, with intensities of 5.9 = 10 19 Wrcm2 , and durations of 7 fs, at a wavelength of 800 nm, are each focused to a 5 mm spot-size. The two beams are offset from each other by 2 mm. The direction of observation is u s 4 rad Žsee Fig. 3.. For each individual electron, Eq. Ž1. gives the velocity zŽ t . as a function of the absolute time t. The acceleration is then calculated, which together with the velocity, is converted to the retarded time tX. For a given direction of observation, Eq. Ž2. predicts the detected field in the laboratory frame. Fig. 3Ža. shows the total electric field of the optical pulse scattered by the
77
electron beam. For the set of parameters used, the scattered pulse is single-cycle, with duration under 0.5 fs. Our simulations show that such short pulses can be generated by those electrons which are accelerated by the optical pulse to velocities above 0.98c. The time lag between the radiated pulses from electrons in different regions of the beam is compensated for using a diffraction grating with a ruling of 3400 linesrmm. For this grating, the time delay between two successive grooves is about 0.7 fs, which exceeds the duration of the pulses radiated by each separate electron. This ensures that the grating performs as a pure delay line Žnot as a spectrally dispersing device.. The corresponding spectrum is shown in Fig. 3Žb.. It is centered close to the 10th harmonic of the pump, i.e. 80 nm. The spectrum is continuous and should result in a single attosecond pulse, not an attosecond pulsetrain. Similar results were obtained using lower peak intensities, lower electron beam energies Ž- 1 keV., or when the position of the electron beam was 300 nm of its optimum position. Thus, the exact experimental conditions are flexible. From a practical viewpoint, recent work on ultrashortpulse Ti:sapphire amplifiers w21,22x has generate terawatt pulses with durations f 20 fs, focusable to intensities ) 10 18 Wrcm2. Even shorter pulses can be generated at lower energies w3x. By combining these technologies, TW pulses with durations under 10 fs can be expected in the near future. Also, to compare the acceleration of an electron in the process of HHG, and the relativistic case
Fig. 3. Ža. Total radiated field, and Žb. spectrum from an electron beam with 100 electrons.
78
I.P. ChristoÕ et al.r Optics Communications 148 (1998) 75–78
discussed herein, we note that due to the stronger field strength utilized here, the acceleration and radiated field is 1000 stronger than in the case of HHG, which is already a useful source of photons. Thus, even with a factor of 10 6 lower electron density, the pulse scattered by the electron beam would be comparable to the intensity of the pulse scattered by an atomic beam with density f 10 18 cmy3, typical of HHG experiments. Taking into account that the HHG intensity is many times weaker than the fundamental scattered light Žin contrast to the relativistic case., even with a low-current electron beam, the conversion efficiency in the relativistic case would be much higher. On the other hand, possible space charge and refractive index effects may limit the upper value of the current of the focused electron beam. Since the technique considered here uses electron beams with diameters up to 1 mm, the total number of electrons may significantly exceed that used in electron lithography. The value of the accessible current depends on the specific design of the electron gun and its focusing optics. In summary, our simulations show that the interaction of a focused electron beam with a femtosecond TW optical pulse can produce single-cycle attosecond pulses in the VUV. If high current density electron guns are used, the expected energy of the attosecond pulse generated by this technique can significantly exceed the energy of attosecond pulses produced by alternate schemes, including highharmonic generation in noble gases. Moreover, the expected scatter beam duration is single cycle in the VUV region of the spectrum.
Acknowledgements The authors gratefully acknowledge support from the National Science Foundation. H.K. acknowledges support from a Sloan Foundation Fellowship.
References w1x J. Zhou, G. Taft, C.P. Huang, M.M. Murnane, H.C. Kapteyn, I. Christov, Optics Lett. 19 Ž1994. 1149. w2x J. Zhou, J. Peatross, M. Murnane, H. Kapteyn, I. Christov, Phys. Rev. Lett 76 Ž1996. 752. w3x M. Nisoli, S.D. Silvestri, O. Svelto, Appl. Phys. Lett. 68 Ž1996. 2793. w4x I.P. Christov, V. Stoev, M. Murnane, H. Kapteyn, Optics Lett. 21 Ž1996. 1493. w5x I.P. Christov, J.P. Zhou, J. Peatross, A. Rundquist, M.M. Murnane, H.C. Kapteyn, Phys. Rev. Lett. 77 Ž1996. 1743. w6x S. Backus, J. Peatross, M. Murnane, H. Kapteyn, Optics Lett. 21 Ž1996. 665. w7x T.W. Hansch, Optics Comm. 80 Ž1990. 71. w8x G. Farkas, C. Toth, Phys. Lett. A 168 Ž1992. 447. w9x P.B. Corkum, N.H. Burnett, M.Y. Ivanov, Optics Lett. 19 Ž1994. 1870. w10x M. Protopapas, D.G. Lappas, C.H. Keitel, P.L. Knight, Phys. Rev. A 53 Ž1996. 2933. w11x P. Antoine, A. L’Huiller, M. Lewenstein, Phys. Rev. Lett. 77 Ž1996. 1234. w12x S.C. Rae, K. Burnett, J. Cooper, Phys. Rev. A 50 Ž1994. 3438. w13x A.E. Kaplan, Phys. Rev. Lett. 73 Ž1994. 1243. w14x S.D. Vartak, N.M. Lawandy, Optics Comm. 120 Ž1995. 184. w15x E.S. Sarachik, G.T. Schappert, Phys. Rev. D 1 Ž1970. 2738. w16x J.H. Eberly, E. Wolf, Progress in Optics, vol. 7, North Holland, Amsterdam, 1969. w17x I.P. Christov, M.B. Danailov, Optics Comm. 68 Ž1988. 291. w18x L. Landau, E. Lifshitz, The Classical Theory of Fields, 2nd ed, ch. 9, Addison-Wesley, Reading, MA, 1951. w19x J.R. Pierce, Theory and Design of Electron Beams, 2nd ed., ch. 6, Van Nostrand, Princeton, 1954. w20x A.E. Siegman, Lasers, University Science Books, Mill Valley, 1986. w21x J.P. Zhou, C.P. Huang, M.M. Murnane, H.C. Kapteyn, Optics Lett. 20 Ž1995. 64. w22x C.P. Barty et al., Optics Lett. 21 Ž1996. 668.