Fundamental constants from photon-photon scattering in three-beam collisions (2024)

Authorization Required

×

×

• 

  Figure 1

  Schematic of the planar three-beam collision. An x-ray beam collides with two optical beams, which are at angles $\mathrm{\Theta }$ and $-\mathrm{\Theta }$ to the counterpropagating direction. (Faint colors show the trajectory of the beams after the collision.)

  Reuse & Permissions

• 

  Figure 2

  Differential number of signal photons $\frac{\mathrm{d}{\mathsf{N}}_{\gamma }}{\mathrm{d}{\theta }_x\mathrm{d}{\theta }_y}$ in three-beam configuration using EuXFEL SASE parameters (see Table1 and the discussion of Sec.5 for more details) with $f^{\#}=1$ optical focusing and an x-ray beam waist $w_{\text{x}}=4\mu \text{m}$. The collision angle is $\mathrm{\Theta }={45}^{\circ }$ and x-ray and optical beams have a relative polarization of $\mathrm{\Delta }\psi =\pi /2$. The angles ${\theta }_x$ and ${\theta }_y$ are the scattering angles of the photons inside and out the interaction plane, respectively [see Eq.(24)]. Dashed vertical lines are at Bragg peak locations [Eq.(25)].

  Reuse & Permissions

• 

  Figure 3

  Logarithmic differential number of signal photons ${log}_{10}[\mathrm{d}{\mathsf{N}}_{\gamma }/\mathrm{d}{\theta }_x\mathrm{d}{\theta }_y]$ in three-beam configuration using future SASE parameters (see Table1 and the discussion of Sec.5 for more details) with $f^{\#}=2$ optical focusing and an x-ray beam waist $w_{\text{x}}=2\mu \text{m}$. The collision angle is $\mathrm{\Theta }={63}^{\circ }$ and x-ray and optical beams have polarization ${\psi }_{\text{x}}={\psi }_0=0$. Dashed vertical lines are at Bragg peak locations [Eq.(25)].

  Reuse & Permissions

• 

  Figure 4

  Polarization dependence of NLO photon-photon scattering. The polarization plane of the x-ray beam is defined by the angle ${\psi }_{\text{x}}$ [see Eq.(15)] and the polarization plane of the optical lasers is defined by ${\psi }_1={\psi }_2={\psi }_0$ [see Eq.(19)]. Shown are the space-time-independent prefactors of (a)${\mathsf{\text{N}}}_{\gamma }^{\parallel }$ and (b)${\mathsf{\text{N}}}_{\gamma }^{\perp }$, given by Eq.(34), and (c)${\mathsf{\text{N}}}_{\gamma }={\mathsf{\text{N}}}_{\gamma }^{\parallel }+{\mathsf{\text{N}}}_{\gamma }^{\perp }$. Each plot has been normalized to a maximum of unity and Eq.(35) has been used.

  Reuse & Permissions

• 

  Figure 5

  Number of signal photons vs collision angle $\mathrm{\Theta }$ using ${\omega }_{\text{x}}=10\mathrm{keV}$ self-seeded (a)EuXFEL parameters and (b)SACLA parameters. Optical pulses have $f^{\#}=1$ focusing, x-ray pulses are focused to $w_{\text{x}}=4\mu \text{m}$, and the relative polarization angle is $\mathrm{\Delta }\psi =\pi /4$. Plotted are the number of photon ${\mathsf{\text{N}}}_{\gamma }^{\parallel }$ scattered into the parallel state (purple solid line), the number ${\mathsf{\text{N}}}_{\gamma }^{\perp }$ scattered into the perpendicular state (blue solid line), and the equivalent photon counts if an angular cut is applied which only accepts photons with emission angles ${\theta }_x>200\mu \text{rad}$ (corresponding dashed lines).

  Reuse & Permissions

• 

  Figure 6

  Estimated lower bound on the number of shots required for $5\sigma$ statistical significance, ${\mathsf{N}}_{\text{shots}}^{5\sigma }=\max ({\mathsf{N}}_{\text{shots}}^{5\sigma ,\parallel },{\mathsf{N}}_{\text{shots}}^{5\sigma ,\perp })$, as a function of the signal-to-background ratio ${\mathsf{N}}_{\gamma ,*}/{\mathsf{N}}_{\gamma }^{\text{bg}}$. The x-ray and optical pulse parameters are as in Fig.5 with $\mathrm{\Theta }={45}^{\circ }$. Estimations are of the number of signal photons held fixed and are given by Table2. The purple solid line shows $f^{\#}=1$ (EuXFEL); purple dashed line, $f^{\#}=2$ (EuXFEL); blue solid line, $f^{\#}=1$ (SACLA); and blue dashed line, $f^{\#}=2$ (SACLA). The data points correspond to the background estimations in Table2.

  Reuse & Permissions

• 

  Figure 7

  Ratio of the number of signal photons in each polarization mode ${\mathsf{\text{N}}}_{\gamma ,*}^{\perp }/{\mathsf{\text{N}}}_{\gamma ,*}^{\parallel }$ vs relative polarization $\mathrm{\Delta }\psi$ for a fixed collision angle $\mathrm{\Theta }={45}^{\circ }$, $f^{\#}=1$ optical focusing, and $w_{\text{x}}=4\mu \text{m}$. The black dashed line shows the analytical result from Eq.(32) with $c_{4,1}$ and $c_{4,2}$ given by Eq.(31), purple circles show the numerically evaluated ratio using EuXFEL parameters, and blue triangles show the numerically evaluated ratio using SACLA parameters.

  Reuse & Permissions

• 

  Figure 8

  Dependence of the number of perpendicular polarized signal photons ${\mathsf{\text{N}}}_{\gamma ,*}^{\perp }$ at collision angle $\mathrm{\Theta }={45}^{\circ }$ on the ratio of the x-ray and optical beam waists $w_{\text{x}}/w_0$ using EuXFEL parameters with $f^{\#}=1$ (purple circles) and $f^{\#}=2$ focusing (blue triangles).

  Reuse & Permissions

• 

  Figure 9

  Number of signal photons vs collision angle $\mathrm{\Theta }$ from the collision of EuXFEL self-seeded ${\omega }_{\text{x}}=10\mathrm{keV}$ photons with two future laser pulses. X-ray pulses are focused to $w_{\text{x}}=4\mu \text{m}$ and optical pulses are focused with $f^{\#}=2$. X-ray and optical pulses have polarization angle $\psi ={\psi }_{★}$ [cf. Eq.(45)], at which the numbers of parallel and perpendicular photons are equal. Plotted are the numbers of parallel ${\mathsf{\text{N}}}_{\gamma ,*}^{\parallel }$ (purple solid line) and perpendicular ${\mathsf{\text{N}}}_{\gamma ,*}^{\perp }$ (blue dashed line) signal photons with emission angles ${\theta }_x>450\mu \text{rad}$.

  Reuse & Permissions

• 

  Figure 10

  Dependence of the signal photons ${\mathsf{\text{N}}}_{\gamma ,*}^{\parallel ,\perp }$ on the polarization $\psi$ for fixed collision angle $\mathrm{\Theta }={63}^{\circ }$. The x-ray and optical pulse parameters are as in Fig.9. Plotted are the number of photons ${\mathsf{\text{N}}}_{\gamma }^{\parallel }$ scattered into the parallel state (purple solid line) and the number ${\mathsf{\text{N}}}_{\gamma }^{\perp }$ scattered into the perpendicular state (blue dashed line), if an angular cut is applied which only accepts photons with emission angles ${\theta }_x>450\mu \text{rad}$.

  Reuse & Permissions

• 

  Figure 11

  Ratio of the number of photons in each polarization mode ${\mathsf{\text{N}}}_{\gamma }^{\perp }/{\mathsf{\text{N}}}_{\gamma }^{\parallel }$ vs polarization $\psi$ for fixed collision angle $\mathrm{\Theta }={63}^{\circ }$. The x-ray and optical pulse parameters as in Fig.9. The black dashed line shows the analytical result from Eq.(37) with $c_{6,1}$ and $c_{6,2}$ given by Eq.(35) and the blue triangles show the numerically evaluated ratio of signal photons with ${\theta }_{\gamma }>{\theta }_{*}$, ${\mathsf{\text{N}}}_{\gamma ,*}^{\perp }/{\mathsf{\text{N}}}_{\gamma ,*}^{\parallel }$.

  Reuse & Permissions

• 

  Figure 12

  Number of signal photons vs collision angle $\mathrm{\Theta }$ in a polarization-insensitive measurement using (a)EuXFEL and (b)SACLA SASE parameters. Optical pulses are focused with $f^{\#}=1$ focusing, x-ray pulses are focused to $w_{\text{x}}=4\mu \text{m}$, and relative polarization angle $\mathrm{\Delta }\psi =\pi /2$. Plotted are the total number of photons ${\mathsf{\text{N}}}_{\gamma }$ (purple solid line) and the equivalent photon counts if an angular cut is applied which only accepts photons with emission angles ${\theta }_x>200\mu \text{rad}$ (blue solid line).

  Reuse & Permissions

• 

  Figure 13

  Dependence of the signal photons ${\mathsf{\text{N}}}_{\gamma ,*}$ on the relative polarization $\mathrm{\Delta }\psi$ for fixed collision angle $\mathrm{\Theta }={45}^{\circ }$ using EuXFEL SASE parameters. Plotted is the total number of photons ${\mathsf{\text{N}}}_{\gamma ,*}$ detected if an angular cut is applied which only accepts photons with emission angles ${\theta }_x>200\mu \text{rad}$ for $f^{\#}=1$ (purple dashed line) and $f^{\#}=2$ (blue dashed line) focusing of the optical laser.

  Reuse & Permissions

• 

  Figure 14

  Estimated lower bound on the number of shots required for $5\sigma$ statistical significance, ${\mathsf{N}}_{\text{shots}}^{\mathsf{n}\sigma }={\mathsf{N}}_{\text{shots}}^{\mathsf{n}\sigma ,\pi /4}+{\mathsf{N}}_{\text{shots}}^{\mathsf{n}\sigma ,\pi /2}$, as a function of the signal-to-background ratio ${\mathsf{N}}_{\gamma ,*}/{\mathsf{N}}_{\gamma }^{\text{bg}}$. The x-ray and optical pulse parameters are as in Fig.12 with $\mathrm{\Theta }={45}^{\circ }$. Estimations of the number of signal photons are held fixed and are given in Table3. The purple solid line shows $f^{\#}=1$ (EuXFEL); purple dashed line, $f^{\#}=2$ (EuXFEL); blue solid line, $f^{\#}=1$ (SACLA); and blue dashed line, $f^{\#}=2$ (SACLA). The data points correspond to the background estimations in Table3.

  Reuse & Permissions

• 

  Figure 15

  Ratio of the total number of scattered photons at different relative polarizations, ${\mathsf{N}}_{\gamma }\left(\mathrm{\Delta }\psi \right)/{\mathsf{N}}_{\gamma }\left(\mathrm{\Delta }{\psi }^{\prime }\right)$. The reference relative polarization is chosen as $\mathrm{\Delta }{\psi }^{\prime }=\pi /2$. The x-ray and optical pulse parameters are as in Fig.12. The black dashed line shows the ratio (33) for $\mathrm{\Delta }{\psi }^{\prime }=\pi /2$, purple circles show the EuXFEL SASE parameters, and blue triangles show the SACLA SASE parameters.

  Reuse & Permissions

• 

  Figure 16

  Relative error $\mathsf{E}=1-{\mathsf{N}}_{\gamma ,*}^{\mathrm{full}}/{\mathsf{N}}_{\gamma ,*}^{\mathrm{IRLA}}$ between the number of signal photons calculated with the full Gaussian pulses in the paraxial approximation, ${\mathsf{N}}_{\gamma ,*}^{\mathrm{full}}$, and with the IRLA, ${\mathsf{N}}_{\gamma ,*}^{\mathrm{IRLA}}$. Also shown are the collision angles at which the number of signal photons is maximized for the currently available parameters (vertical dashed) and future parameters (vertical dotted).

  Reuse & Permissions

×