High Harmonic Generation

Albert Liu University of Michigan, Ann Arbor

ince the invention of the maser, doped liquid droplets with high intensity lasers, and subsequent advent of the laser, which unsuprisingly faces many issues such as Sthe demands of both industry and producing debris that damage the focusing optics academia have pushed for the extension of after a few uses [8]. This problem in particular efficient coherent light generation towards reduces the practicality of the method due to the shorter wavelengths. Problems have be- expensive cost of ultra-violet optics. The lack of come apparent however, in the direct gen- practical methods for EUV generation has thus eration of extreme-ultraviolet (EUV) to x- fueled intense study into alternative methods of ray wavelengths and beyond. Generation generation that is supported by government as by non-linear optical processes have there- well as industry. fore become a very attractive and practical solution, of which high-harmonic generation (HHG) has become a leading con- Background tender. Traditionally, the polarization P of a material seemed linearly related to an externally applied Motivation field: P = 0χE (1) There are many motivations existing in both Under applied intense light however, the polar- industry and academia for desiring short wave- ization is revealed to have nonlinear components: length radiation. The most evident need for EUV (1) (2) 2 (3) 3 and shorter wavelength radiation is in the inte- P = 0 χ E + χ E + χ E + ... (2) grated circuits market, in which new lithographic techniques are necessary to push towards the In 1961 Peter Franken and others at the Uni- sub-50nm feature sizes of the future. Extreme versity of Michigan first observed harmonic gen- ultra-violet lithography is a potential technique eration, in which light of a given frequency ω that may reach 32nm sizes [1], but the generation is used to produce additional light of frequency EUV radiation is an outstanding technological nω, where n is a positive integer. Specifically, barrier towards practical implementation. The they directed a ruby laser onto a quartz sample current most promising EUV generation tech- in which the intense coherent light resulted in nique for lithography application is pulsing tin- second harmonic generation (n = 2), also known

Page 1 of 6 as frequency doubling [6]. This effectively began account the quantum nature of atoms but leaves the entire field of non-linear optics, in which the the incident radiation as classical fields, and a nonlinear components of the polarization are ex- fully quantum view that quantizes both the atoms ploited to produce many spectacular processes. and the radiation fields. After Franken’s groundbreaking result, there were no breakthroughs for the next 16 years; much Semi-Classical Picture work went into studying the frequency doubling 3 The semi-classical view was first proposed by Paul process and a fractional harmonic n = 2 was observed. In 1977 however, the first harmonic Corkum in 1993, which is colloquially referred line above the third harmonic (n = 3) was finally to as the ”two step model”. In this model, a observed by Burnett and colleagues. In fact, by strong incident field ionizes a valence electron pulsing solid aluminum targets with 20-50J of from an atom which tunnels into the continuum. power they observed the generation of frequencies The probability of ionization can be calculated as high as the eleventh harmonic [2]. This natu- using Fermi’s golden rule, but after tunneling the rally raised the question: is there a limit to the electron is treated classically. The free electron harmonic order able to be produced? Today, the is then considered to appear outside of the atom shortest wavelengths that have been produced by while effectively at rest, but the applied field harmonic generation exceed the 300th harmonic. immediately accelerates the electron away from its parent nucleus. In one half an optical period, the electron begins to accelerate back towards the nucleus and eventually recombines or ”impact ionizes” - releasing a train of harmonics:

Figure 1: A timeline of the highest harmonics gen- Figure 2: ”Two-Step Model” of HHG, in which a erated by year. The CO2 laser genera- valence electron is ionized by an intense tion occurred with solid generation me- field then recombines to release a har- dia, whereas the Nd : glass/Y AG, KrF , monic train and T i : Al2O3 laser generation used rare gases [4]. Although the model has the drawback of leav- ing the recombination mechanism unspecified, it For perspective, a typical Ti:Sapph laser sys- does produce the observed experimental spectra tem which emits 800nm radiation can be used to quite well - especially the cutoff in the highest generate photons of energies greater than 500eV harmonic able to be generated. We first define (λ < 2.5nm). The production of harmonics above the ponderomotive potential Up, which is the clas- the fifth is referred to as high harmonic genera- sical kinetic energy a charged particle will gain tion. in an oscillating electromagnetic field:

e2E2 U = 0 (3) Theory p 4mω2 The physical origin of HHG is still an area of If the calculation is performed in calculating the active research. In general however, there have velocity distribution of the accelerating electron, arisen two main frameworks to explain the phe- we ﬁnd that the most probable energy it will nomena: a semi-classical view that takes into have when nearing the nucleus will be 3.17Up.

Page 2 of 6 The energy of the maximum harmonic able to be ﬁlled with an ionized argon plasma at very high produced will thus be: pressure [12]:

Emax = Ip + 3.17Up (4) where Ip is the energy needed to ionize the electron.

Fully Quantum Picture Although the full quantum theory of HHG is too involved for detailed examination in this re- Figure 3: Spectra from HHG induced in multiply view, we will note a few key features. In 1998, ionized Argon (Ar5+). The blue spectra Gao and others proposed an alternative, fully is low incident intensity optimized for UV quantum physical picture to that offered by the generation while the purple is high inten- semi-classical model. In the fully quantum pic- sity optimized for x-ray generation. ture, the intense optical field incident on an atom couples to a valence electron, forming a station- the notable result from this experiment was the ary state called a Volkov state. To emphasize, high conversion efficiency (incident pulse photon the Volkov state is a coupled electron-photon to harmonic photon ratio) exhibited by the plas- −3 −7 system that satisfies Schrodinger’s equation and mas - 10 to 10 across the EUV to soft x-ray can be thought of as the electron and photon regimes, which are orders of magnitude greater together forming an atomic bound state. The than in traditional noble gas HHG. emission of radiation is therefore attributed to transitions between Volkov states, which has the Ellipticity Dependence interesting property of preserving the momen- There is a strong dependence of HHG on the tum of the electron [7]. The generation rate for ellipticity of the incident light, defined by the each harmonic can therefore be calculated which, ratio of its relative components Ey . Corkum and Ex along with the predicted photon polarization and colleagues first showed that the intensity of the harmonic spectral distributions, agree quite well harmonics generated decreases with increasing with experimental data. ellipticity, plotted in Figure 4:

HHG Data

We now discuss some real experimental high harmonic spectra that have been observed in experi- ments. In the past, high harmonic generation was achieved primarily with solid generation media. More recently however, the dominant method has been to use pressurized noble gases such as Argon or Neon. In an effort to reach higher harmonic generation limits and efficiencies, recent experi- ments have begun pushing towards HHG in more exotic media such as ionized plasmas. Shown in Figure 3 is a spectrum presented in a very recent (December 4, 2015) issue of Science by Figure 4: Intensity dependence of the (a) 41st har- Kapteyn and Murnane’s group at JILA, in which monic from Ne, and (b) 21st harmonic a Ti:Sapphire laser of pulse energy 2.6mJ and from Ar with respect to driving field ellip- pulse duration 35fs was directed onto a waveguide ticity (plotted with normalized units).

Page 3 of 6 The reason for this dependence can be most eas- practical application of HHG. In this section, we ily justfied in the semi-classical model. As the list two more applications of HHG to emphasize ellipticity of the incident field increases, the path the impact of the technique. of the ionized electron will increasingly deviate from a straight line away from the parent nuclei Attosecond Pulse Generation and back. Any deviation in the electronic path will affect the probability of recombination of the In modern physics, there is always the need to electron with its parent nuclei decreases. As seen probe ever-shorter timescales. Though common- in the plot, the intensity maximum exists at zero place now, the breakthroughs in femtosecond ellipticity which corresponds to linearly polarized pulse generation allowed scientists to probe molec- light [5]. ular dynamics - a timescale regime that once seemed unattainable. Today, there is yet another Intensity Dependence threshold that may be probed with shorter pulse lengths. That is, the attosecond timescales of In the typical HHG experiment, the intensity electron dynamics. There was much excitement of the harmonics will increase corresponding to in the burgeoning stages of HHG research due to increasing pulse intensity. Shown in Figure 5 is apparent observations of harmonic pulse widths the intensity dependence of a single harmonic: shorter than the femtosecond pulses that generated them. For a long time the exact pulse durations were unable to be measured, but in 2001 Paul and colleagues successfully measured the timescales of the harmonic pulse trains emit- ted during HHG. Astonishingly, the pulse widths were observed to be in the attosecond time-scale:

Figure 5: Intensity dependence of the 25th harmonic from Argon with respect to the driving pulse intensity (plotted with normalized units) [3].

16 W At intensities approaching I = 5 ∗ 10 cm2 however, the electric dipole approximation that ig- nores the magnetic contribution in the Lorentz force begins to break down. In the semi-classical model, the magnetic component of the Lorentz force begins to exert an eﬀect on the electronic motion, causing it to curve away from an other- Figure 6: HHG from an Argon gas generation wise linear path - in the same way as elliptically medium was observed to emit a harmonic pulse train of pulse widths approximately polarized light does [14]. This thus limits the in- 250 attoseconds, separated by 1.35 fem- tensity of the driving laser, and thus the highest toseconds. Note that these values repre- harmonic achievable. sent averages, since each pulse will diﬀer in their temporal characteristics by small Applications of HHG amounts [11].

In the motivation section, we listed the generation Also, methods to select a single pulse from the of EUV/X-ray radiation for lithographic use as a harmonic train have long been proposed, which

Page 4 of 6 makes HHG an attractive method for probing the minimums that will cause the reconstruction to unexplored attosecond regime. stagnate. A schematic diagram of the algorithm is presented in Figure 7: Coherent Diffractive Imaging In the last few decades, numerous techniques have been developed to image micro-scale and smaller feature sizes such as electron microscopy, atomic force microscopy etc. Each of these techniques presents unique problems however, such as limited area of view or thickness limits on the sample. A technique that is impervious to many of these issues is a relatively new technique called ”coherent diffractive imaging” (CDI). First ex- Figure 7: Schematic of the ER algorithm. perimentally demonstrated in 1999 by Miao and colleagues [10], traditional CDI has 3 essential CDI is otherwise known as ”lensless imaging”, parts: since lenses are not fundamentally needed. This makes CDI a truly diffraction limited technique, 1. Coherent light source incident on sample to which means shorter wavelength radiation di- produce diffraction pattern. rectly correspond to higher imaging resolution. X-rays for example offer the potential to image 2. Detector to record diffraction pattern. at nano-scale resolution, without the limitations 3. Computer algorithm to retrieve the sample posed by other techniques of comparable resolving image from its diffraction pattern. power. The main challenge of practical implementation is the generation of such x-rays. In A computer algorithm is necessary due to the days past one would have needed to travel to well-known phase problem - first identified in the large synchrotron facilities for coherent x-rays, crystallography field. The problem is because of but fortunately HHG offers a table-top source of the lack of phase information in the diffraction coherent x-ray photons. An example HHG EUV pattern recorded (intensity is dependent upon CDI image is shown: the amplitude squared). In fact, the phase often carries more information about the sample image than the amplitude itself. Fortunately, iterative algorithms have been developed to ”reconstruct” the sample image from its diffraction pattern alone, the simplest of which is the error-reduction (ER) algorithm. Essentially, the algorithm first makes a guess for the sample image (amplitude and phase). This guess is then Fourier transformed to find its potential far-field diffraction pattern, which will of course be completely wrong. The amplitudes of this guessed diffraction pattern will be replaced with the recorded diffraction pattern amplitudes, and will then be inverse Fourier transformed back to find a new, more accurate image. This represents one iteration in the algo- Figure 8: a) SEM image of sample. b) Recorded rithm, and is repeated for an arbitrary number diffraction pattern. c) Magnitude of re- of times until the desired accuracy is reached for constructed image. d) Intensity profile the reconstructed image. This method is prone of the green line drawn in c), showing a to many issues however, such as local transform spatial resolution of 214nm [13].

Page 5 of 6 Moreover, HHG x-ray sources have the further [7] J. Gao, F. Shen, and J.G. Eden, ”Quantum advantage of short pulse time durations down to Electrodynamic Treatment of Harmonic Gen- the attosecond range; this gives CDI both spatial eration in Intense Optical Fields” Physics and temporal resolution simultaneously. Review Letters, 81, 1833 (1998).

[8] George et al., ”13.5nm EUV Generation Conclusion From Tin-Doped Droplets Using A Fiber Laser” Opt. Express, 15, 16348-16356 In this review, we have explained the current (2007). theoretical framework behind high harmonic generation, presented the typical characteristics of [9] Ivanov and Corkum, ”Routes to Control of harmonics produced by the method, and demon- Intense-Field Atomic Polarizability” Physics strated a few of the numerous applications for Review Letters, 74-15, 2933-2936 (1995). HHG. Essentially, the two main characteristics [10] Miao et al., ”Extending the Methodology of HHG - generation of short wavelength radia- of X-ray Crystallography To Allow Imaging tion and sub-femtosecond pulse production - are of Micrometre-Sized Non-Crystalline Speci- what make the technique invaluable to so many mens” Nature, 400, 342-344 (1999). areas of future research. As the HHG technique is reﬁned and perfected in the next few years, [11] Paul et al., ”Observation of a Train of At- expect the technique to become commonplace in tosecond Pulses From High Harmonic Gen- laboratories across the globe. eration” Science, 292, 1689-1694 (2001).

[12] Popmintchev et al., ”Ultraviolet Surprise: References Eﬃcient Soft X-Ray High-Harmonic Gener- ation in Multiply Ionized Plasmas” Science, [1] H. Bennett, ”Will Future Measurement 350-6265, 1225-1231 (2015). Needs of the Semiconductor Industry Be Met?” J. Res. Natl. Inst. Stand. Technol., [13] Sandberg et al., ”Lensless Diﬀractive Imag- 112, 25-38 (2007). ing Using Tabletop Coherent High-Harmonic Soft X-Ray Beams” Physics Review Letters, [2] Burnett et al., ”Harmonic Generation in 399, (2015). CO2 Laser Target Interaction” Applied Physics Letters, 31, 172 (1977). [14] Walser et al., ”High Harmonic Generation [3] Dinh, Hannaford, and Dao, ”Intensity De- Beyond the Electric Dipole Approximation” pendent Spectral Features in High Harmonic Physical Review Letters, 85-24, 5082-5085 Generation” Journal of Applied Physics, (2000). 113, (2013). [4] J.G. Eden, ”High-order Haronic Genera- tion and Other Intense Optical Field-Matter Interactions: Review of Recent Experimen- tal and Theoretical Advances” Progress in Quantum Electronics, 28, 197-246 (2004). [5] Dietrich et al., ”High Harmonic Generation and Correlated Two-Electron Multiphoton Ionization With Elliptically Polarized Light” Physical Review A, 50-5, (1994). [6] Franken et al., ”Generation of Optical Har- monics” Physics Review Letters, 7-4, 118- 119 (1961).

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