Model & Assumptions Beta

Model: Special-relativistic kinematics: K = (γ − 1)m_ec², γ = (1 − v²/c²)^−1/2, λ_e = h/p.
Assumptions: Free electron in vacuum. Constants from CODATA 2018.
Limitations: No potential energy or interaction effects. The de Broglie wavelength is the free-space value — inside a material, effective wavelength differs.

Module

Parameters

v / c

K (keV)

λ_e (pm)

v (nm/fs)

E (J)

Quick set

Field probe

R (nm)

z (nm)

Spectral scan

ℏω_min (eV)

ℏω_max (eV)

Velocity —

Relativistic kinematics. K = (γ − 1) m_ec², γ = (1 − v²/c²)^−½, λ_e = h / p. Constants: CODATA 2018.

λ_e vs K —

FWHM of |E(t)| —

Frequency-domain field —

Time-domain field —

References

[1]F. J. García de Abajo, Optical excitations in electron microscopy, Reviews of Modern Physics 82, 209–275 (2010), doi:10.1103/RevModPhys.82.209.
[2]J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Parameters

v / c

K (keV)

Spacing a (nm)

Impact para. b (nm)

Number of elements N

Steering mode ξ

Dipole orientation

Induced dipole direction

comp Re Im

p_x

p_y

p_z

|p| = 1.000

Azimuth φ (°)

Emission angle θ (°)

Wavelength (nm)

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⤢ enlarge

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Steering mode — induced dipole profile ξ = 0

Notes

Generalized Smith–Purcell. A swift electron (v/c, energy K) flies parallel to a periodic array of N point emitters (period a, impact parameter b) and induces a coherent dipole on each. Each Fourier mode ℓ of the (engineered) dipole profile opens an emission channel at sin θ_n,ℓ = 1/β − (n − ℓ/N) λ/a (Eq. 6); ℓ = 0 is the ordinary Smith–Purcell line. Steering ξ modulates |p_j| = 1 + sin(2πξj/N) to select channel ℓ = ξ.

Polarization. The electron field drives an elliptical dipole d = (i/γ)K₀ x̂ + K₁ ẑ (Eq. 9): the i makes x and z oscillate 90° out of phase. In the φ = 0 plane the emission is purely p-polarized; off-plane (φ ≠ 0) it becomes elliptical, with wavelength- and angle-dependent handedness (Stokes S₃).

References

[1]E. J. C. Dias, A. Rodríguez Echarri, T. P. Rasmussen, F. J. García de Abajo & J. D. Cox, Active steering of cathodoluminescence through a generalized Smith–Purcell effect, Light: Science & Applications 15, 218 (2026), doi:10.1038/s41377-026-02280-y.
[2]S. J. Smith & E. M. Purcell, Visible Light from Localized Surface Charges Moving across a Grating, Phys. Rev. 92, 1069 (1953), doi:10.1103/PhysRev.92.1069.

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Spectral axis unit

Map each quantity to a file column (1-based). Unmapped components are taken as zero. The swift electron's near-field has no y-component, so α_y does not drive emission in this geometry. Only the spectral shape of α matters — its overall magnitude, units, and global phase cancel in the normalized SP outputs.

Relativistic Electron Calculator

Parameters

References

Parameters

Notes

References

Confirm polarizability input α(ω)

Parameters

Notes

References