Phase-Matching Technique for High-Harmonic Generation
Unusual approach may provide an alternative to x-ray lasers.
Researchers at the University of Colorado at Boulder have shown how to accomplish quasi-phasematching in ionized gases, thereby enabling enhancements by two orders of magnitude in the conversion efficiency of high-order harmonic generation. They believe that the technique may lead to desktop sources of coherent x-ray beams, providing a very practical alternative to huge and expensive x-ray lasers. A practical source of coherent radiation in the x-ray region would have many important applications, including high-resolution imaging and advanced lithography.
Figure 1. The researchers directed a pulse from a Ti:sapphire laser through a waveguide filled with argon gas, where the intense pulse generated high-order harmonic radiation. They phase-matched the process by counterpropagating a short Ti:sapphire pulse train through the waveguide. The counterpropagating pulses suppressed harmonic generation periodically along the length of the waveguide so that harmonic radiation was generated only from regions that added coherently to produce the extreme-ultraviolet output beam. Images reprinted with permission of Nature Physics.
Experimentally, the researchers directed a femtosecond pulse from a Ti:sapphire laser through a hollow waveguide containing argon gas (Figure 1). Inside the waveguide, they pulse ionized the gas, and extreme ultraviolet radiation at ~18 nm (70 eV photon energy, the 43rd harmonic of the 770-nm fundamental) was generated when the temporarily freed electrons slammed back into their ions.
This effect has been demonstrated in many laboratories. What is new in the Colorado work is that the researchers were able to phase-match the harmonic-generation process at these photon energies so that the extreme ultraviolet radiation emerged in a bright, coherent, laserlike beam.
A green laser pointer is probably the most familiar example of phase-matched harmonic generation. For decades, phase-matching has been achieved in nonlinear crystals by harnessing the crystal’s birefringence to compensate for dispersion between the fundamental and harmonic electromagnetic waves.
More recently, quasi-phase-matching has been successfully employed in many applications and has found its way into numerous commercial products. In this technique, the fundamental and harmonic waves move at different phase velocities in the nonlinear crystal and become out of phase with each other after propagating a short distance. But the crystal is periodically poled along the direction of propagation so that, as soon as the waves become out of phase, one of them is flipped by 180° so they are back in phase.
These phase-matching techniques work fine for wavelengths down to hundreds of nanometers, but at shorter wavelengths, the photons become so energetic that propagation in crystals is impossible. Ionized inert gases have become the medium of choice for the generation of harmonics in the extreme-ultraviolet and x-ray spectral regions. But gases are not birefringent, and they cannot be periodically poled, so phase-matching high-harmonic generation in ionized gases has become one of the “grand challenges” in modern photonics.
Now the researchers in Colorado believe that they have met this challenge by employing a technique somewhat analogous to quasi-phase-matching in crystals. Rather than periodically flipping the phase of one of the waves, they suppress harmonic generation from those regions that would generate out-of-phase harmonic radiation. The result is the same in either case: The harmonic radiation emerges in a laserlike, coherent beam.
Simplistically, as a fundamental wave propagates into a nonphase-matched nonlinear crystal, the harmonic generated in the first picometer (call that point A) is in phase with the harmonic generated in the second picometer. But as the fundamental propagates many micro- or millimeters into the crystal, it eventually reaches point B, where, as a result of dispersion between the fundamental and harmonic waves, the harmonic generated at point B is exactly out of phase with the harmonic generated at point A. Therein lies the problem solved by phase-matching.
In a quasi-phase-matched crystal, the crystal’s polarization is periodically reversed (the crystal is “periodically poled”) so that the phase between the fundamental and the harmonic is shifted by 180° at the polarization boundary. Thus, the harmonic generated at point B is exactly in phase with the harmonic generated in the first few picometers. Likewise, subsequent reversals of the crystal’s polarization ensure that the harmonic waves generated at points C, D and so on are all in phase.
The new quasi-phase-matching technique in gases is different. In this case, a counterpropagating pulse (or pulse train) quenches the harmonic generated at point B (and at points D, F and so on). Harmonic signal is preferentially generated in those regions of the gas where it adds in phase to produce a coherent output beam.
The counterpropagating pulses periodically quench the harmonic generated by the forward-propagating pulse because they interfere with the forward-moving field, modulating both its phase and amplitude. By choosing the experimental parameters judiciously, the researchers translated this modulation into the periodic modulation of the harmonic efficiency they required for quasi-phase-matching (Figure 2).
Figure 2. A weak counterpropagating pulse (as shown in Figure 1) modulates the intensity of harmonic generation spatially along the direction of propagation in the nonlinear medium. This graphic shows the spatial intensities of the 37th through 45th harmonics. For quasi-phase-matching to occur, the harmonic signals generated at the peaks must all be in phase; the harmonic generated in the valleys is out of phase with the harmonic generated at the peaks.
In one of their most incisive demonstrations, the researchers boosted the 43rd harmonic (70-eV photons) by a factor of more than 300 when they counterpropagated a pulse train of three weak pulses through the waveguide. They believe that this result, which produced a flux of 1010 photons per second, clearly demonstrates that the new phase-matching technique is effective in ionized gases where phase-matching has previously not been possible. In the future, they hope to extend their results to shorter wavelengths, well into the soft x-ray region and perhaps even for hard x-ray generation, and also to obtain further efficiency increases.
Nature Physics, online Feb. 25, 2007, doi:10.1038/nphys541.
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