# Coherent, Short-Pulse X-ray Generation via Relativistic Flying Mirrors

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory of Relativistic Mirrors

^{′}) denotes the variables in the rest frame K and $\gamma ={(1-{\beta}^{2})}^{-1/2}$ is the relativistic factor of the mirror. The light phase $\varphi =\omega t-\mathbf{k}\xb7\mathbf{r}$ is Lorentz invariant, where $\mathbf{k}$ is the wave vector of the light and $\mathbf{r}$ is the position vector. We obtain

^{′}, the angle of reflection is same as that of incidence and the frequency does not change; thus we obtain ${\alpha}^{\prime}=\pi -{\theta}^{\prime}$, ${\omega}_{r}^{\prime}={\omega}^{\prime}$. Finally, we return to the laboratory frame by the inverse Lorentz transformation and obtain

## 3. Several Implementations of Relativistic Flying Mirrors

#### 3.1. Relativistic Charged Beam

#### 3.2. Propagating Ionization Front

#### 3.3. Moving Boundary of Impedance in Nonlinear Transform Line

#### 3.4. Moving Boundary of Electron-Hole Plasma in Semiconductors

#### 3.5. Oscillating Mirror/Sliding Mirror

^{20}W/cm

^{2}intensity laser pulse with a double plasma mirror [24]. They observed up to 238th harmonic of the initial laser frequency where the spectrum decays with a power law as expected by the theory. Later Dromey et al. observed nearly diffraction limited harmonics radiation in the wavelength of 20–40 nm [25].

#### 3.6. A Thin Foil Mirror Driven by an Intense Laser Light Pressure

^{2}and another weak laser pulse (∼2 mJ, 55 fs) was focused onto the opposite side of the target at the intensity of $1\times {10}^{15}$ W/cm

^{2}. Frequencies of 8th to 15th harmonics of the fundamental laser frequency were observed as shown in Figure 2. The wavelengths of the reflected signal ranged from 50 nm to 100 nm. The upshift factor was ∼10 and the reflectivity of the mirror was estimated to be 5 × ${10}^{-5}$ in terms of photon number.

#### 3.7. Breaking Wake Waves

#### 3.8. Superluminal Mirrors

## 4. Applications of Relativistic Flying Mirrors

## 5. Conclusions

^{−5}in terms of photon number but more systematic measurement is demanded. In addition, an increase of the reflected photon number is critical for practical applications for ultrafast imaging, etc.

## Acknowledgments

## Conflicts of Interest

## References

- Einstein, A. Zur Elektrodynamik bewegter Körper. Annalen Phys.
**1905**, 322, 891–921. [Google Scholar] [CrossRef] - Kando, M. Light reflection from mirror moving at relativistic speed—Reflection and frequency-upshifting pf electromagnetic waves from wake waves produced through laser-plasma interaction. J. Plasma Fusion Res.
**2010**, 86, 164–168. [Google Scholar] - Bulanov, S.V.; Esirkepov, T.Z.; Kando, M.; Pirozhkov, A.S.; Rosanov, N.N. Relativistic mirrors in plasmas. Novel results and perspectives. Phys. Usp.
**2013**, 56, 429–464. [Google Scholar] [CrossRef] - Motz, H. Applications of the Radiation from Fast Electron Beams. J. Appl. Phys.
**1951**, 22, 527–535. [Google Scholar] [CrossRef] - Granatstein, V.L.; Sprangle, P.; Parker, R.K.; Pasour, J.; Herndon, M.; Schlesinger, S.P.; Seftor, J.L. Realization of a relativistic mirror: Electromagnetic backscattering from the front of a magnetized relativistic electron beam. Phys. Rev. A
**1976**, 14, 1194–1201. [Google Scholar] [CrossRef] - Huang, Z.; Kim, K.J. Review of x-ray free-electron laser theory. Phys. Rev. Accel. Beams
**2007**, 10, 034801. [Google Scholar] [CrossRef] - Sasao, N. Intense gamma radiation by accelerated quantum ions. In Proceedings of the Light driven Nuclear-Particle Physics and Cosmology LNPC, Yokohama, Japan, 19–21 April 2017; pp. 1–19. [Google Scholar]
- Semenova, V. Reflection of electromagnetic waves from an ionization front. Radiophys. Quantum Electron.
**1967**, 10, 599–604. [Google Scholar] [CrossRef] - Kuo, S. Frequency up-conversion of microwave pulse in a rapidly growing plasma. Phys. Rev. Lett.
**1990**, 65, 1000–1003. [Google Scholar] [CrossRef] [PubMed] - Savage, R.; Joshi, C.; Mori, W. Frequency up-conversion of electromagnetic-radiation upon transmission into an ionization front. Phys. Rev. Lett.
**1992**, 68, 946–949. [Google Scholar] [CrossRef] [PubMed] - Mori, W. Generation of tunable radiation using an underdense ionization front. Phys. Rev. A
**1991**, 44, 5118–5121. [Google Scholar] [CrossRef] [PubMed] - Lai, C.; Liou, R.; Katsouleas, T.C.; Muggli, P.; Brogle, R.; Joshi, C.; Mori, W.B. Demonstration of microwave generation from a static field by a relativistic ionization front in a capacitor array. Phys. Rev. Lett.
**1996**, 77, 4764–4767. [Google Scholar] [CrossRef] [PubMed] - Yugami, N.; Fujita, K.; Higashiguchi, T.; Nishida, Y. Experimental observation of short microwave generation via relativistic ionization front produced by CO
_{2}laser. Jpn. J. Appl. Phys.**1998**, 37, 688–689. [Google Scholar] [CrossRef] - Higashiguchi, T.; Yugami, N.; Okabe, H.; Niiyama, T.; Takahashi, E.; Ito, H.; Nishida, Y. Emission of Short Microwave Pulse Radiated by Interaction between Periodic Static Electric Field and Relativistic Ionization Front. Jpn. J. Appl. Phys.
**1999**, 38, L527–L530. [Google Scholar] [CrossRef] - Itoh, H.; Soda, K. Doppler Frequency Shift of an Electromagnetic Wave by a Fast Moving Boundary. OYO BUTURI
**1999**, 48, 616–622. [Google Scholar] - Mu, L.; Donaldson, W.R.; Adams, J.C.; Falk, R.A. Electromagnetic wave interaction with laser-induced plasmas in GaAs. In Optically Activated Switching IV; Donaldson, W.R., Ed.; International Society for Optics and Photonics: Bellingham, WA, USA, 1995; pp. 107–113. [Google Scholar]
- Bae, J.; Xian, Y.J.; Yamada, S.; Ishikawa, R. Doppler frequency up conversion of electromagnetic waves in a slotline on an optically excited silicon substrate. Appl. Phys. Lett.
**2009**, 94, 091120. [Google Scholar] [CrossRef] - Kohno, N.; Itakura, R.; Tsubouchi, M. Mechanism of relativistic Doppler reflection from a photoinduced moving plasma front studied by terahertz time-domain spectroscopy. Phys. Rev. B
**2016**, 94, 155205. [Google Scholar] [CrossRef] - Bulanov, S.V.; Naumova, N.; Pegoraro, F. Interaction of an ultrashort, relativistically strong laser-pulse with an overdense plasma. Phys. Plasmas
**1994**, 1, 745–757. [Google Scholar] [CrossRef] - Naumova, N.; Nees, J.; Sokolov, I.; Hou, B.; Mourou, G. Relativistic generation of isolated attosecond pulses in a λ
^{3}focal volume. Phys. Rev. Lett.**2004**, 92, 063902. [Google Scholar] [CrossRef] [PubMed] - Gordienko, S.; Pukhov, A.; Shorokhov, O.; Baeva, T. Relativistic Doppler effect: Universal spectra and zeptosecond pulses. Phys. Rev. Lett.
**2004**, 93, 115002. [Google Scholar] [CrossRef] [PubMed] - Pirozhkov, A.S.; Bulanov, S.V.; Esirkepov, T.Z.; Mori, M.; Sagisaka, A.; Daido, H. Attosecond pulse generation in the relativistic regime of the laser-foil interaction: The sliding mirror model. Phys. Plasmas
**2006**, 13, 013107. [Google Scholar] [CrossRef] - Pirozhkov, A.S.; Bulanov, S.V.; Esirkepov, T.Z.; Mori, M.; Sagisaka, A.; Daido, H. Generation of high-energy attosecond pulses by the relativistic-irradiance short laser pulse interacting with a thin foil. Phys. Lett. A
**2006**, 349, 256–263. [Google Scholar] [CrossRef] - Dromey, B.; Zepf, M.; Gopal, A.; Lancaster, K.; Wei, M.S.; Krushelnick, K.; Tatarakis, M.; Vakakis, N.; Moustaizis, S.; Kodama, R.; et al. High harmonic generation in the relativistic limit. Nat. Phys.
**2006**, 2, 456–459. [Google Scholar] [CrossRef] - Dromey, B.; Adams, D.; Hoerlein, R.; Nomura, Y.; Rykovanov, S.G.; Carroll, D.C.; Foster, P.S.; Kar, S.; Markey, K.; McKenna, P.; et al. Diffraction-limited performance and focusing of high harmonics from relativistic plasmas. Nat. Phys.
**2009**, 5, 146–152. [Google Scholar] [CrossRef] - Nomura, Y.; Hoerlein, R.; Tzallas, P.; Dromey, B.; Rykovanov, S.; Major, Z.; Osterhoff, J.; Karsch, S.; Veisz, L.; Zepf, M.; et al. Attosecond phase locking of harmonics emitted from laser-produced plasmas. Nat. Phys.
**2009**, 5, 124–128. [Google Scholar] [CrossRef] - Vincenti, H.; Quere, F. Attosecond Lighthouses: How to Use Spatiotemporally Coupled Light Fields to Generate Isolated Attosecond Pulses. Phys. Rev. Lett.
**2012**, 108, 113904. [Google Scholar] [CrossRef] [PubMed] - Esirkepov, T.Z.; Bulanov, S.; Kando, M.; Pirozhkov, A.S.; Zhidkov, A.G. Boosted High-Harmonics Pulse from a Double-Sided Relativistic Mirror. Phys. Rev. Lett.
**2009**, 103, 025002. [Google Scholar] [CrossRef] [PubMed] - Kulagin, V.V.; Cherepenin, V.A.; Hur, M.S.; Suk, H. Flying mirror model for interaction of a super-intense nonadiabatic laser pulse with a thin plasma layer: Dynamics of electrons in a linearly polarized external field. Phys. Plasmas
**2007**, 14, 113101. [Google Scholar] [CrossRef] - Bulanov, S.S.; Maksimchuk, A.; Krushelnick, K.; Popov, K.I.; Bychenkov, V.Y.; Rozmus, W. Ensemble of ultra-high intensity attosecond pulses from laser–plasma interaction. Phys. Lett. A
**2010**, 374, 476–480. [Google Scholar] [CrossRef] - Kiefer, D.; Yeung, M.; Dzelzainis, T.; Foster, P.S.; Rykovanov, S.G.; Lewis, C.L.; Marjoribanks, R.S.; Ruhl, H.; Habs, D.; Schreiber, J.; et al. Relativistic electron mirrors from nanoscale foils for coherent frequency upshift to the extreme ultraviolet. Nat. Commun.
**2013**, 4, 1763. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bulanov, S.; Esirkepov, T.Z.; Tajima, T. Light intensification towards the Schwinger limit. Phys. Rev. Lett.
**2003**, 91, 085001. [Google Scholar] [CrossRef] [PubMed] - Kando, M.; Fukuda, Y.; Pirozhkov, A.S.; Ma, J.; Daito, I.; Chen, L.; Esirkepov, T.Z.; Ogura, K.; Homma, T.; Hayashi, Y.; et al. Demonstration of laser-frequency upshift by electron-density modulations in a plasma wakefield. Phys. Rev. Lett.
**2007**, 99, 135001. [Google Scholar] [CrossRef] [PubMed] - Pirozhkov, A.; Ma, J.; Kando, M.; Esirkepov, T.Z.; Fukuda, Y.; Chen, L.M.; Daito, I.; Ogura, K.; Homma, T.; Hayashi, Y.; et al. Frequency multiplication of light back-reflected from a relativistic wake wave. Phys. Plasmas
**2007**, 14, 123106. [Google Scholar] [CrossRef] - Panchenko, A.V.; Esirkepov, T.Z.; Pirozhkov, A.S.; Kando, M.; Kamenets, F.F.; Bulanov, S.V. Interaction of electromagnetic waves with caustics in plasma flows. Phys. Rev. E
**2008**, 78, 056402. [Google Scholar] [CrossRef] [PubMed] - Bulanov, S.V.; Esirkepov, T.Z.; Kando, M.; Koga, J.K.; Pirozhkov, A.S.; Nakamura, T.; Bulanov, S.S.; Schroeder, C.B.; Esarey, E.; Califano, F.; et al. On the breaking of a plasma wave in a thermal plasma. II. Electromagnetic wave interaction with the breaking plasma wave. Phys. Plasmas
**2012**, 19, 113103. [Google Scholar] [CrossRef] - Kando, M.; Pirozhkov, A.S.; Kawase, K.; Esirkepov, T.Z.; Fukuda, Y.; Kiriyama, H.; Okada, H.; Daito, I.; Kameshima, T.; Hayashi, Y.; et al. Enhancement of Photon Number Reflected by the Relativistic Flying Mirror. Phys. Rev. Lett.
**2009**, 103, 235003. [Google Scholar] [CrossRef] [PubMed] - Pirozhkov, A.S.; Kando, M.; Esirkepov, T.Z.; Fukuda, Y.; Chen, L.M.; Daito, I.; Ogura, K.; Homma, T.; Hayashi, Y.; Kotaki, H.; et al. Demonstration of Flying Mirror with Improved Efficiency. AIP Conf Proc.
**2009**, 1153, 274–284. [Google Scholar] - Lampe, M.; Ott, E.; Waler, J.H. Interaction of electromagnetic waves with a moving ionization front. Phys. Fluids
**1978**, 21, 42–54. [Google Scholar] [CrossRef] - Bu, Z.; Shen, B.; Huang, S.; Li, S.; Zhang, H. Light reversing and folding based on a superluminal flying mirror in a plasma with increasing density. Plasma Phys. Control. Fusion
**2016**, 58, 1–9. [Google Scholar] [CrossRef] - Hashimshony, D.; Zigler, A.; Papadopoulos, K. Conversion of Electrostatic to Electromagnetic Waves by Superluminous Ionization Fronts. Phys. Rev. Lett.
**2001**, 86, 2806–2809. [Google Scholar] [CrossRef] [PubMed] - Vshivkov, V.A.; Naumova, N.M.; Pegoraro, F.; Bulanov, S.V. Nonlinear electrodynamics of the interaction of ultra-intense laser pulses with a thin foil. Phys. Plasmas
**1998**, 5, 2727–2741. [Google Scholar] [CrossRef] - Pirozhkov, A.S.; Esirkepov, T.Z.; Pikuz, T.A.; Faenov, A.Y.; Ogura, K.; Hayashi, Y.; Kotaki, H.; Ragozin, E.N.; Neely, D.; Kiriyama, H.; et al. Burst intensification by singularity emitting radiation in multi-stream flows. Sci. Rep.
**2017**, 7, 17968. [Google Scholar] [CrossRef] [PubMed] - Saldin, E.L.; Schneidmiller, E.A.; Shvyd’ko, Y.V.; Yurkov, M.V. XFEL with a meV Bandwidth: Seeding Option; TESLA Technical Design Report; Deutschen Elektronen-Synchrotron Notkestrasse: Hamburg, Germany, 2001; Part V, Additional Material on CD-Rom; p. 332. [Google Scholar]
- Koga, J.K.; Bulanov, S.V.; Esirkepov, T.Z.; Pirozhkov, A.S.; Kando, M.; Rosanov, N.N. Possibility of measuring photon-photon scattering via relativistic mirrors. Phys. Rev. A
**2012**, 86, 053823. [Google Scholar] [CrossRef] - Bulanov, S.S.; Maksimchuk, A.; Schroeder, C.B.; Zhidkov, A.G.; Esarey, E.; Leemans, W.P. Relativistic spherical plasma waves. Phys. Plasmas
**2012**, 19, 020702. [Google Scholar] [CrossRef] - Chen, P.; Mourou, G. Accelerating Plasma Mirrors to Investigate the Black Hole Information Loss Paradox. Phys. Rev. Lett.
**2017**, 118, 045001. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The light reflection by an inclined flying mirror. (

**a**,

**c**) are shown in the laboratory frame and (

**b**) is in the mirror rest frame. $\mathbf{k}$ is the wave vector and prime (′) denotes the variables in the rest frame.

**Figure 2.**Experimentally observed spectra reproduced from [31]. (

**a**,

**b**) are spectra from the shots without counter-propagating pulses while (

**c**,

**d**) are spectra with them. (

**e**) Detector image obtained from a 50-nm probe shot.

**Figure 3.**Relativistic flying mirrors of breaking plasma waves showing one dimensional (

**a**) and three dimensional (

**b**) representations.

**Figure 4.**Signal intensity distribution obtained in the experiment. $\Delta t$ and $\Delta z$ denote the time and vertical position differences between the two laser pulses.

**Figure 5.**Reflected signals reproduced from [37]. (

**a**) Raw charge-coupled device (CCD) image after the transmission grating. (

**b**) Spectra with the diffraction orders of +1 and −1. (

**c**) CCD counts within the 1st diffraction order vs. time delay between the driver and source pulses; also shown are results of the shots without the source pulse the delays of which are assinged arbitrarily.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kando, M.; Esirkepov, T.Z.; Koga, J.K.; Pirozhkov, A.S.; Bulanov, S.V. Coherent, Short-Pulse X-ray Generation via Relativistic Flying Mirrors. *Quantum Beam Sci.* **2018**, *2*, 9.
https://doi.org/10.3390/qubs2020009

**AMA Style**

Kando M, Esirkepov TZ, Koga JK, Pirozhkov AS, Bulanov SV. Coherent, Short-Pulse X-ray Generation via Relativistic Flying Mirrors. *Quantum Beam Science*. 2018; 2(2):9.
https://doi.org/10.3390/qubs2020009

**Chicago/Turabian Style**

Kando, Masaki, Timur Zh. Esirkepov, James K. Koga, Alexander S. Pirozhkov, and Sergei V. Bulanov. 2018. "Coherent, Short-Pulse X-ray Generation via Relativistic Flying Mirrors" *Quantum Beam Science* 2, no. 2: 9.
https://doi.org/10.3390/qubs2020009