# 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

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**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.

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**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