Interface

@field(name) do ... end

Frontend macro for creating an electric field and storing it in the variable name.

Example

julia> @field(F) do
           I₀ = 2.0
           T = 2.0
           σ = 3.0
           Tmax = 3.0
       end
Linearly polarized field with
  - I₀ = 2.0000e+00 au = 7.0188904e16 W cm⁻² =>
    - E₀ = 1.4142e+00 au = 727.2178 GV m⁻¹
    - A₀ = 0.4502 au
  – a Fixed carrier @ λ = 14.5033 nm (T = 48.3777 as, ω = 3.1416 Ha = 85.4871 eV)
  – and a Gaussian envelope of duration 170.8811 as (intensity FWHM; ±2.00σ)

field_amplitude(f, t)

Compute the field amplitude from the vector_potential $A(t)$ using automatic differentiation according to

\[F(t) = -\partial_t A(t).\]

field_amplitude(f, a, b)

Compute the time integral of the field amplitude according to

\[\int_a^b\diff{t}F(t) = -[A(b) - A(a)],\]

where $A(t)$ is the vector_potential.

vector_potential(f, t)

Compute the vector potential $\vec{A}(t)$ of the field f.

instantaneous_intensity(f, t)

Compute the instantaneous intensity of the field f as

\[I_i(t) = \abs{F(t)}^2 = \abs{-\partial_t A(t)}^2.\]

intensity(f, t)

Compute the time-dependent intensity of the field f by phase-shifting the carrier continuously until the maximum instantaneous_intensity is achieved.

field_envelope(f, t)

Compute the field amplitude envelope as the square root of intensity.

span(f)

Return the end-points for the envelope of the field f.

span(f::ApodizedField)

The span of f is given by the intersection of span(parent(f)) and the apodizing interval a .. b.

timeaxis(f[, fs])

Construct a time axis for the field f, covering the span of the envelope in the number of steps given by the sample frequency fs.

timeaxis(f, (N,dt))

Alternate interface that constructs the time axis with N steps starting from first(span(f)), with intervals of dt.

steps(f[, fs])

Return number of time steps for the field f, optionally supplying a sampling frequency.

We construct the time axis such that the highest frequency component of the field is resolved. By the Nyquist sampling theorem, we need minimum $f_s>2f_{\textrm{max}}$, but to be on the safe side, we use, as default, $f_s=100f_{\textrm{max}}$. This also makes plots nicer.

If fs<:Integer, the sampling frequency is computed as fs/T, i.e. fs steps per cycle.

spectrum(env::AbstractGaussianEnvelope)

Gaussians belong to the Schwartz class, i.e. functions who, under Fourier transform, are mapped back to the same space. That is to say, the Fourier transform of a Gaussian is a Gaussian:

\[\exp(-\alpha t^2) \leftrightarrow \frac{1}{\sqrt{2\alpha}} \exp\left(-\frac{\omega^2}{4\alpha}\right).\]

Comparing with the above, we find that the spectral standard deviation

\[\Omega = \sqrt{2\alpha} = \frac{2\sqrt{\ln 2}}{\tau},\]

and the Gaussian function in the spectral domain is thus

\[E(\omega) = \frac{E_0\tau}{2\sqrt{\ln 2}} \exp\left[-\frac{(\omega\tau)^2}{8\ln2}\right].\]

field_amplitude_spectrum(f::AbstractField, ω)

Compute the analytic Fourier transform of the field_amplitude of the field f at the angular frequency ω, using the Fourier identity

\[\vec{F}(t) = -\partial_t \vec{A}(t) \iff \hat{\vec{F}}(\omega) = -\im\omega \hat{\vec{A}}(\omega)\]

vector_potential_spectrum(f::AbstractField, ω)

Compute the analytic Fourier transform of the vector_potential of the field f at the angular frequency ω.

fft(f::AbstractField, t::AbstractRange)

Compute the FFT of the field_amplitude of the field f sampled on the uniform temporal grid t.

nfft(y, ts...)

Normalized FFT of y with respect to the axes ts. We use the symmetric normalization $(2\pi)^{-N/2}$ where $N$ is the number of dimensions.

nfft(f::AbstractField, t::AbstractRange)

Compute the symmetrically normalized FFT of the field_amplitude of the field f sampled on the uniform temporal grid t.

fft_vector_potential(f::AbstractField, t::AbstractRange)

Compute the FFT of the vector_potential of the field f sampled on the uniform temporal grid t.

nfft_vector_potential(f::AbstractField, t::AbstractRange)

Compute the symmetrically normalized FFT of the vector_potential of the field f sampled on the uniform temporal grid t.

fftω(t::AbstractRange)

Return the angular frequency grid corresponding to uniform sampling in time with the tempolar grid t.

rfft(f::AbstractField, t::AbstractRange)

Compute the RFFT of the field_amplitude of the field f sampled on the uniform temporal grid t, in the case of real-valued signals.

irfft(F, t)

Compute the IRFFT of the spectral amplitudes F, with the end-result resolved on the time axis t, in the case of real-valued signals.

rfft_vector_potential(f::AbstractField, t::AbstractRange)

Compute the RFFT of the vector_potential of the field f sampled on the uniform temporal grid t, in the case of real-valued signals.

rfftω(t::AbstractRange)

Return the angular frequency grid corresponding to uniform sampling in time with the tempolar grid t, in the case of real-valued signals.