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Figure 1.9: Schematic of the intuitive three-step model.

Figure 1 9: Schematic of the intuitive three-step model.

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Figure 1.1: The first reports on high order harmonic generation. (A) A plot of the relative energy-conversion efficiency for harmonic generation in Ne shows a sharp change in slope between the ninth and eleventh harmonics. The typical error bar is shown for the seventh harmonic. Adapted from [7]. (B) The relative intensity of the harmonics generated in Xe, Kr and Ar. The typical error bar is shown for the 15th harmonic. Adapted from [8]. Abstract quote: ’The key point is that the harmonic intensity falls slowly beyond the fifth harmonic as the order increases.’
Figure 1.1: The first reports on high order harmonic generation. (A) A plot of the relative energy-conversion efficiency for harmonic generation in Ne shows a sharp change in slope between the ninth and eleventh harmonics. The typical error bar is shown for the seventh harmonic. Adapted from [7]. (B) The relative intensity of the harmonics generated in Xe, Kr and Ar. The typical error bar is shown for the 15th harmonic. Adapted from [8]. Abstract quote: ’The key point is that the harmonic intensity falls slowly beyond the fifth harmonic as the order increases.’
Figure 1.11: Normalized electron wavepacket calculated at the peak of a three cycle laser pulse of different wavelengths. The laser polarization is parallel to the longer spatial extent.
Figure 1.11: Normalized electron wavepacket calculated at the peak of a three cycle laser pulse of different wavelengths. The laser polarization is parallel to the longer spatial extent.
Figure 1.16: High harmonics are usually generated in a free focusing geometry (gas jet) (A), or in a waveguide geometry (hollow waveguide (B) and plasma waveguide (D)). If higher laser intensities are available, one can use also propagation in long Rayleigh length geometry (loosely focused laser beam, C), relative to the interaction region length, where the geometric dispersion can be neglected. Note the differences in the possible extension of the medium length relative to the Rayleigh range length. Tight focus geometries (i.e. HHG in gas jets) in general lead to higher divergence and lower spatial coherence of the generated X-ray beams compared to guided geometries (i.e. HHG in hollow waveguides, etc.).
Figure 1.16: High harmonics are usually generated in a free focusing geometry (gas jet) (A), or in a waveguide geometry (hollow waveguide (B) and plasma waveguide (D)). If higher laser intensities are available, one can use also propagation in long Rayleigh length geometry (loosely focused laser beam, C), relative to the interaction region length, where the geometric dispersion can be neglected. Note the differences in the possible extension of the medium length relative to the Rayleigh range length. Tight focus geometries (i.e. HHG in gas jets) in general lead to higher divergence and lower spatial coherence of the generated X-ray beams compared to guided geometries (i.e. HHG in hollow waveguides, etc.).
Table 1.2: Coefficients u,,; for the lower guided modes in the waveguide dispersion relation 1.60 (the I*” zero of the Bessel function Jm—1(um) = 0).
Table 1.2: Coefficients u,,; for the lower guided modes in the waveguide dispersion relation 1.60 (the I*” zero of the Bessel function Jm—1(um) = 0).
Figure 1.18: Measurement of the coherence of phase matched high harmonic generation in a hollow waveguide. Measurements were made with two pinholes at different sepa- rations. Notice the very high fringe visibility out to almost 80% width of the beam. Adapted from [14]. Figure 1.18: Measurement of the coherence of phase matched high harmonic generation
Figure 1.18: Measurement of the coherence of phase matched high harmonic generation in a hollow waveguide. Measurements were made with two pinholes at different sepa- rations. Notice the very high fringe visibility out to almost 80% width of the beam. Adapted from [14]. Figure 1.18: Measurement of the coherence of phase matched high harmonic generation
Figure 1.21: Evolution of the QPM techniques. (C) Mechanical approaches at low photon energies - modulated hollow waveguides [16, 17] and sequences of gas jets [18]. (B) All-optical approach at low photon energies - using counterpropagating pulse train (19]. (A) All-optical approach in the keV photon energy range - grating-assisted phase matching [20] (see Chapter 3).
Figure 1.21: Evolution of the QPM techniques. (C) Mechanical approaches at low photon energies - modulated hollow waveguides [16, 17] and sequences of gas jets [18]. (B) All-optical approach at low photon energies - using counterpropagating pulse train (19]. (A) All-optical approach in the keV photon energy range - grating-assisted phase matching [20] (see Chapter 3).
Figure 1.23: Intensity of the 29” harmonic generated from Ar for various ionization evels as a function of gas pressure in the plane wave case. Absorption is artificially set to zero, a = 0. The left column shows the case for ionization levels much higher than the optimum value, for pressures between 0 and 1 torr: 100% (a), 50% (b), and 10% (c). The right column shows the case for levels close to the optimum value of the ionization or pressures between 0 and 40 torr: 4.0% (d), 3.8% (e), and 3.5% (f). The signal levels on the right are 3 to 6 orders of magnitude larger than those on the left. Adapted from 22]
Figure 1.23: Intensity of the 29” harmonic generated from Ar for various ionization evels as a function of gas pressure in the plane wave case. Absorption is artificially set to zero, a = 0. The left column shows the case for ionization levels much higher than the optimum value, for pressures between 0 and 1 torr: 100% (a), 50% (b), and 10% (c). The right column shows the case for levels close to the optimum value of the ionization or pressures between 0 and 40 torr: 4.0% (d), 3.8% (e), and 3.5% (f). The signal levels on the right are 3 to 6 orders of magnitude larger than those on the left. Adapted from 22]
Figure 2.8: HHG optimization. Harmonic emission from 6 Torr of Ar using 11 mJ, 28 fs laser pulses and various peak discharge currents: 5 A (solid red), 10 A (dotted blue), 15 A (dash-dot green), and no discharge (dashed black). Each spectrum was measured at a delay of r=2 pus. The HHG signal decreases at lower harmonic orders and remains the same at high harmonic orders. The increasing electron density with current leads to further reduction of the coherence lengths.
Figure 2.8: HHG optimization. Harmonic emission from 6 Torr of Ar using 11 mJ, 28 fs laser pulses and various peak discharge currents: 5 A (solid red), 10 A (dotted blue), 15 A (dash-dot green), and no discharge (dashed black). Each spectrum was measured at a delay of r=2 pus. The HHG signal decreases at lower harmonic orders and remains the same at high harmonic orders. The increasing electron density with current leads to further reduction of the coherence lengths.
Figure 2.12: (a) Spectra showing the blueshift of 9 mJ laser pulses for various currents in the capillary discharge initially filled with 5.2 Torr Ar. The spectra for discharge currents of 0, 10, 20, and 30 A, and through an evacuated capillary are shown with central wavelengths (vertical bars) of 773, 778, 784, 787, and 793 nm, respectively. (b) High harmonic spectra, from a 9 mJ pulse propagating through 5.8 Torr Ar, with and without the discharge present, showing the effects of blueshifting of the fundamental spectrum. Spectra for 0, 10, 20, and 25 A are shown, as is the location of the center of the 47") harmonic (vertical bars).
Figure 2.12: (a) Spectra showing the blueshift of 9 mJ laser pulses for various currents in the capillary discharge initially filled with 5.2 Torr Ar. The spectra for discharge currents of 0, 10, 20, and 30 A, and through an evacuated capillary are shown with central wavelengths (vertical bars) of 773, 778, 784, 787, and 793 nm, respectively. (b) High harmonic spectra, from a 9 mJ pulse propagating through 5.8 Torr Ar, with and without the discharge present, showing the effects of blueshifting of the fundamental spectrum. Spectra for 0, 10, 20, and 25 A are shown, as is the location of the center of the 47") harmonic (vertical bars).
Figure 2.13: Continuous tuning of the wavelength of individual harmonics (zoom in). These data show a total shift of more than two harmonic orders (i.e. approximately 3 eV).
Figure 2.13: Continuous tuning of the wavelength of individual harmonics (zoom in). These data show a total shift of more than two harmonic orders (i.e. approximately 3 eV).
Figure 2.15: Near field images of intense laser beam in the exit plane of the capillary at different time delays during the current pulse in Fig. 2.14. The initial Ar pressure was 6.0 Torr and 10 A current pulses were used. The input laser pulses were 28 fs FWHM in duration and had energies of 8.5 mJ. different time delays during the current pulse in Fig. 2.14. ‘The initial Ar pressure was
Figure 2.15: Near field images of intense laser beam in the exit plane of the capillary at different time delays during the current pulse in Fig. 2.14. The initial Ar pressure was 6.0 Torr and 10 A current pulses were used. The input laser pulses were 28 fs FWHM in duration and had energies of 8.5 mJ. different time delays during the current pulse in Fig. 2.14. ‘The initial Ar pressure was
Figure 2.16: Comparison of near field images of low intensity (< 10!4 W/cm?) laser pulses exiting the discharge at (a) 200 ns before the onset of the current pulse, (b) 600 ns, and (c) 1200 ns after the onset of the current pulse. The discharge was operated under the same conditions as in Fig. 2.15.
Figure 2.16: Comparison of near field images of low intensity (< 10!4 W/cm?) laser pulses exiting the discharge at (a) 200 ns before the onset of the current pulse, (b) 600 ns, and (c) 1200 ns after the onset of the current pulse. The discharge was operated under the same conditions as in Fig. 2.15.
Figure 3.1: Schematic of high-order difference-frequency mixing (HDM) in plasma waveguides
Figure 3.1: Schematic of high-order difference-frequency mixing (HDM) in plasma waveguides
Table 3.1: Effective conversion efficiencies (the generated power normalized by the power generated for perfect phase matching) of QPM in low-order nonlinear processes vesrus that of GAPM in high-order harmonic generation. respectively [104]. is given by (2/m7)? which for the first four orders are 0.405, 0.101, 0.045, and 0.025,
Table 3.1: Effective conversion efficiencies (the generated power normalized by the power generated for perfect phase matching) of QPM in low-order nonlinear processes vesrus that of GAPM in high-order harmonic generation. respectively [104]. is given by (2/m7)? which for the first four orders are 0.405, 0.101, 0.045, and 0.025,
Figure 3.7: Comparison between all-optical QPM (top) and GAPM (bottom) in high- order harmonic generation.
Figure 3.7: Comparison between all-optical QPM (top) and GAPM (bottom) in high- order harmonic generation.
Figure 3.8: GAPM using multicolor counterpropagating quasi-cw waves. The plots present the total HHG phase and the growth of the HHG field along the propagation. (top) Bicolor counterpropagating beam which component fields induce phase A®y#cG that consists of the first two terms of the Fourier series of a saw-tooth function that are at fundamental frequency and its second harmonic. The HHG signal becomes stronger as additional terms of the Fourier series are added, asymptotically approaching perfect phase matching. (bottom) If there is a phase offset of 7/2 (the sine wave changes to a cosine wave), the Fourier series corresponds to a Shah function (sequence of delta functions), resulting in lower HHG signal. In the case of a single counterpropagating wave, the phase offset does not affect the HHG signal buildup.
Figure 3.8: GAPM using multicolor counterpropagating quasi-cw waves. The plots present the total HHG phase and the growth of the HHG field along the propagation. (top) Bicolor counterpropagating beam which component fields induce phase A®y#cG that consists of the first two terms of the Fourier series of a saw-tooth function that are at fundamental frequency and its second harmonic. The HHG signal becomes stronger as additional terms of the Fourier series are added, asymptotically approaching perfect phase matching. (bottom) If there is a phase offset of 7/2 (the sine wave changes to a cosine wave), the Fourier series corresponds to a Shah function (sequence of delta functions), resulting in lower HHG signal. In the case of a single counterpropagating wave, the phase offset does not affect the HHG signal buildup.
Figure 3.11: GAPM in standing-wave light structure, using same color driving and counterpropagating light A; = 0.8 um (black (a), red (b)), versus GAPM in traveling- wave light structure (blue, for the same parameters as in Fig. 3.10 (a)). The pressure for case of GAPM in standing-wave light structure is increased from 70 torr in (a) to 140 torr in (b) to show also the 2”4-order GAPM.
Figure 3.11: GAPM in standing-wave light structure, using same color driving and counterpropagating light A; = 0.8 um (black (a), red (b)), versus GAPM in traveling- wave light structure (blue, for the same parameters as in Fig. 3.10 (a)). The pressure for case of GAPM in standing-wave light structure is increased from 70 torr in (a) to 140 torr in (b) to show also the 2”4-order GAPM.
10" ph/sec 10’ph/sec 10’ph/sec 100 ph/sec
10" ph/sec 10’ph/sec 10’ph/sec 100 ph/sec
Figure 4.4: Direct verification of perfect phase matching of HHG at Ay, = 0.8 wm by measuring the coherence length L,,, = a/Ak of a specific harmonic using counter- propagating light. Adapted from [23]. The normalized intensity modulations of 27” generated in Ar versus pulse collision point and pressure are directly correlated with the coherence length and disappear under perfect phase matching conditions (/Ak = 0) at 45 torr. Inset: Unnormalized harmonic intensity as a function of pressure demonstrating perfect phase matching at 45 torr.
Figure 4.4: Direct verification of perfect phase matching of HHG at Ay, = 0.8 wm by measuring the coherence length L,,, = a/Ak of a specific harmonic using counter- propagating light. Adapted from [23]. The normalized intensity modulations of 27” generated in Ar versus pulse collision point and pressure are directly correlated with the coherence length and disappear under perfect phase matching conditions (/Ak = 0) at 45 torr. Inset: Unnormalized harmonic intensity as a function of pressure demonstrating perfect phase matching at 45 torr.
Figure 4.8: (A) Experimental macroscopic HHG signal growth versus pressure at pho- ton energies close to the phase matching cutoffs for 1.3 um lasers. (B) Theoretical predictions of the pressure required for phase matching using 0.8 and 1.3 wm lasers, based on a simple semi-analytical model (solid lines). (B) Quantum calculations for He including 1D propagation.
Figure 4.8: (A) Experimental macroscopic HHG signal growth versus pressure at pho- ton energies close to the phase matching cutoffs for 1.3 um lasers. (B) Theoretical predictions of the pressure required for phase matching using 0.8 and 1.3 wm lasers, based on a simple semi-analytical model (solid lines). (B) Quantum calculations for He including 1D propagation.
Figure 4.138: Flux comparison. Experimental phase-matched HHG emission from He, Ne (Ag/Zr filters - solid/dashed line), and Ar driven by A,=1.3 yum, illustrating the sharp HHG cutoffs ((A) double logarithmic scale, (B) linear scale).
Figure 4.138: Flux comparison. Experimental phase-matched HHG emission from He, Ne (Ag/Zr filters - solid/dashed line), and Ar driven by A,=1.3 yum, illustrating the sharp HHG cutoffs ((A) double logarithmic scale, (B) linear scale).
Figure 4.14: Spatial domain. Raw image of the near-perfect Gaussian, low-divergence, soft X-ray beam generated in He, for a broad band of photon energies (200-330 eV), spanning into the water window.
Figure 4.14: Spatial domain. Raw image of the near-perfect Gaussian, low-divergence, soft X-ray beam generated in He, for a broad band of photon energies (200-330 eV), spanning into the water window.
Figure 4.17: Scaling of the pressure-length products required for efficient phase-matched HHG emission at the phase matching cutoffs for laser wavelengths up to A, =6 wm and corresponding phase matching limits up to 3 keV. (A) Predicted optimal pressure show- ing quadratic growth with A,. (B) Absorption-limited medium length for reaching 90% of the asymptotic limit of the macroscopic harmonic signal, determined by reabsorption.
Figure 4.17: Scaling of the pressure-length products required for efficient phase-matched HHG emission at the phase matching cutoffs for laser wavelengths up to A, =6 wm and corresponding phase matching limits up to 3 keV. (A) Predicted optimal pressure show- ing quadratic growth with A,. (B) Absorption-limited medium length for reaching 90% of the asymptotic limit of the macroscopic harmonic signal, determined by reabsorption.
Figure 4.19: Experimental setup. Legend: CPA I-II - chirped pulse amplification stages of Ti:Sp laser, WLG - white light generation, OPA [-II-III - optical parametric amplifi- cation, p - pump beam, cwl - chirped white light, s - signal beam, i - idler beam.
Figure 4.19: Experimental setup. Legend: CPA I-II - chirped pulse amplification stages of Ti:Sp laser, WLG - white light generation, OPA [-II-III - optical parametric amplifi- cation, p - pump beam, cwl - chirped white light, s - signal beam, i - idler beam.
Figure 4.21: Differential pumping apparatus designed to reduce the pressure inside the vacuum chamber following the exit of the hollow waveguide. Typically pressures of about 10000 torr inside the waveguide decrease down to few percents of a torr after the device. Figure 4.21: Differential pumping apparatus designed to reduce the pressure inside the
Figure 4.21: Differential pumping apparatus designed to reduce the pressure inside the vacuum chamber following the exit of the hollow waveguide. Typically pressures of about 10000 torr inside the waveguide decrease down to few percents of a torr after the device. Figure 4.21: Differential pumping apparatus designed to reduce the pressure inside the
ADK). the dependence of the ionization rate on the ionization potential of the gas (assuming gas species, the small variation of the power law scaling of wg(A,) is a consequence o
ADK). the dependence of the ionization rate on the ionization potential of the gas (assuming gas species, the small variation of the power law scaling of wg(A,) is a consequence o
Hence, the group velocity of the driving laser is always less than the group velocity dispersion all the terms in dnz/dX have the same sign: group velocity matching vg, = ¢ cannot be achieved since for a medium with norma
Hence, the group velocity of the driving laser is always less than the group velocity dispersion all the terms in dnz/dX have the same sign: group velocity matching vg, = ¢ cannot be achieved since for a medium with norma
Figure 4.25: (A) Schematic of the drift of the electron wavepacket due to the magnetic component of the laser field, along the propagation direction away from the nucleus (0,0). (B) Squared ratio, ¢, of the drift of the wavepacket and its spread versus Xz, showing that magnetic field effects may decrease the HHG flux only for Ay approaching 6 wm in He and 7 wm in Ne. These laser wavelengths correspond to full phase matching cutoffs of about 3 keV (point (1) and (2) in Fig. 4.6 A or Fig. 4.16)
Figure 4.25: (A) Schematic of the drift of the electron wavepacket due to the magnetic component of the laser field, along the propagation direction away from the nucleus (0,0). (B) Squared ratio, ¢, of the drift of the wavepacket and its spread versus Xz, showing that magnetic field effects may decrease the HHG flux only for Ay approaching 6 wm in He and 7 wm in Ne. These laser wavelengths correspond to full phase matching cutoffs of about 3 keV (point (1) and (2) in Fig. 4.6 A or Fig. 4.16)
Table A.1: Ionization potentials J,, of selected atoms and their ions [5]
Table A.1: Ionization potentials J,, of selected atoms and their ions [5]
Table C.1: Absorption edges of noble gas atoms [6].
Table C.1: Absorption edges of noble gas atoms [6].
Related topics:
Nonlinear OpticsHigh harmonic generation
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