Lasers
EPOCH can support laser sources on the boundaries, with a wide array of spatial and temporal profiles. These examples provide basic input decks for simulating lasers in vacuum. General documentation on the EPOCH laser block can be found here.
Simple plane wave
This input deck provides the most basic form of laser. We use uniform spatial and temporal profiles, and inspect the Poynting flux in the simulation window. As a typical rule of thumb, allow 20 cells per wavelength to get a good resolution.
begin:control
nx = 200
ny = 400
t_end = 100e-15
x_min = 0
x_max = 10e-6
y_min = -10e-6
y_max = 10e-6
stdout_frequency = 100
end:control
begin:boundaries
bc_x_min = simple_laser
bc_x_max = open
bc_y_min = open
bc_y_max = open
end:boundaries
begin:laser
boundary = x_min
intensity_w_cm2 = 1.0e18
lambda = 1.0e-6
end:laser
begin:output
dt_snapshot = 10 * femto
poynt_flux = always
end:output
As we have specified a time-averaged intensity of $10^{18} \text{ Wcm}^{-2}$, we expect the Poytning flux to range 0 to $2\times 10^{18} \text{ Wcm}^{-2}$, which we can see here. Note that, because our laser profile reaches the $y_{min}$ and $y_{max}$ boundaries, some noise has been introduced here. In practice, these boundary effects are less of an issue, as simulations are expected to mainly deal with laser pulses.
Laser pulse
Using the maths parser, we can create a laser pulse which has a Gaussian profile in time and space for the intensity distribution, $I$:
$ I(y,t) \propto e^{-(y-y_0)^2/2\sigma_y^2} e^{-(t-t_0)^2/2\sigma_t^2}$
where we consider a 2D simulation with an $x$ propagating laser.
Let the full-width-at-half-maximum of the spatial and temporal profiles be 5 $\mu m$ and 40 fs respectively for the intensity $I$ distribution. The relationship between $\sigma$ and the $fwhm$ for a Gaussian distribution is:
$\sigma = \frac{fwhm}{2\sqrt{2 \ln(2)}}$
However, the profile keys in the laser block describe modifications to the boundary electric fields, $E$, and as $I \propto E^2$, the $E$ profile must satisfy
$ E(y,t) \propto e^{-((y-y_0)/2\sigma_y)^2} e^{-((t-t_0)/2\sigma_t)^2}$
The EPOCH
maths parser provides
a special gauss(x,x_0,w) command for use in the input.deck, which sets a
profile of the form:
$e^{-(x-x_0)^2/w^2}$
Hence, we may use this gauss function to model the electric field profiles if
we set
$w = \frac{fwhm}{\sqrt{2\ln(2)}}$
where $fwhm$ refers to the intensity distribution.
We can let the spatial profile peak at $y=0$. We don’t want the laser to peak at $t=0$, as this would ignore the rising intensity. Instead, let us start when the laser pulse is 10% of its maximum value - the half-width-at-10%-maximum, $hw0.1m$. For a Gaussian beam, this is:
$hw0.1m = \frac{fwhm}{2} \sqrt{\frac{\ln(10)}{\ln(2)}}$
begin:control
nx = 700
ny = 400
t_end = 100e-15
x_min = 0
x_max = 35e-6
y_min = -10e-6
y_max = 10e-6
stdout_frequency = 100
end:control
begin:boundaries
bc_x_min = simple_laser
bc_x_max = open
bc_y_min = open
bc_y_max = open
end:boundaries
begin:constant
t_fwhm = 40.0e-15
y_fwhm = 5.0e-6
w_t = t_fwhm / sqrt(2*loge(2))
w_y = y_fwhm / sqrt(2*loge(2))
t_hw01m = 0.5 * t_fwhm * sqrt(loge(10)/loge(2))
end:constant
begin:laser
boundary = x_min
intensity_w_cm2 = 1.0e18
lambda = 1.0e-6
profile = gauss(y,0,w_y)
t_profile = gauss(time,t_hw01m,w_t)
end:laser
begin:output
dt_snapshot = 10 * femto
poynt_flux = always
end:output
Here we see that the laser $fwhm$ in $x$ and $y$ are 12 $\mu m$ and 5 $\mu m$ respectively, where the $x$ $fwhm$ corresponds to a temporal FWHM of 40 fs, as expected. Also, because there is little contact between the pulse and the boundaries, we do not have any numerical boundary disturbance.
Focussing a Gaussian Beam
A laser can be driven on the boundary so that it focusses on a given spot. Basic details of how to do this are here. To summarise, using the paraxial approximation, the electric fields for a $x$-propagating, $y$-polarised Gaussian beam take the form:
$\pmb{E}(r,x) = E_0 \frac{w_0}{w(x)} e^{-r^2/w(x)^2} e^{-i(kx + k\frac{r^2}{2R_c(x)}-\psi(x))} \hat{\pmb{y}}$
where
- $r$ is the radial distance from the laser propagation axis
- $x$ is axial distance along the wave, with $x=0$ at the focus
- $E_0$ is the peak electric field amplitude at the focus
- $w(x)$ is the beam-waist at $x$ (radial distance where field strength drops by $e^{-1}$)
- $w_0$ is $w(x=0)$
- $k$ is the laser wave-vector
- $R_c(x)$ is the radius of curvature at $x$
- $\psi(x)$ is the Gouy phase correction
If the fields on the simulation boundary are of this form, then the fields will propagate according to this equation, and a focal spot will be formed. Note that this propagation is only expected provided the paraxial approximation is satisfied. This implies that, for vacuum propagation, the laser wavelength, $\lambda$ is much smaller than the beam-waist: $\lambda « w_0$.
The following deck gives an example for a laser attached to x_min. Two constant blocks are provided: the first gives the user control over the focused laser properties, and the second derives variables to be used in the laser block. The user only needs to touch the first, which sets the intensity full-width-at-half-maximum (related to beam-waist), the peak, cycle-averaged intensity, the laser wave-length and the distance from the $x_{min}$ boundary to the focal point.
begin:control
nx = 2400
ny = 1200
t_end = 100e-15
x_min = 0
x_max = 20e-6
y_min = -5e-6
y_max = 5e-6
stdout_frequency = 100
end:control
begin:boundaries
bc_x_min = simple_laser
bc_x_max = open
bc_y_min = open
bc_y_max = open
end:boundaries
begin:constant
I_fwhm = 2.0e-6 # FWHM of laser intensity
I_peak_Wcm2 = 1.0e15 # 0.5 * eps0 * c * E_peak^2
las_lambda = 1.0e-6 # Laser wavelength
foc_dist = 5.0e-6 # Boundary to focal point distance
end:constant
begin:constant
las_k = 2.0 * pi / las_lambda
w0 = I_fwhm / sqrt(2.0 * loge(2.0)) # Beam Waist
ray_rang = pi * w0^2 / las_lambda # Rayleigh range
w_boundary = w0 * sqrt(1.0 + (foc_dist/ray_rang)^2) # Waist on boundary
I_boundary = I_peak_Wcm2 * (w0 / w_boundary)^2 # Intens. on boundary
rad_curve = foc_dist * (1.0 + (ray_rang/foc_dist)^2) # Boundary curv. rad.
gouy = atan(-foc_dist/rad_curve) # Boundary Gouy shift
end:constant
begin:laser
boundary = x_min
intensity_w_cm2 = I_boundary
lambda = las_lambda
phase = las_k * y^2 / (2.0 * rad_curve) - gouy
profile = gauss(y, 0, w_boundary)
end:laser
begin:output
name = o1
dt_snapshot = 10 * femto
poynt_flux = always
end:output
In this example, EPOCH correctly reproduces the focal point position, laser wavelength, and radial FWHM at the focus - however, the peak intensity is only $0.88\times 10^{15} \text{ Wcm}^{-2}$. This intensity reduction from target is due to the tight focal spot, with $w_0\approx 1.7$ μm being close to $\lambda = 1.0$ μm.
The deck is based on the laser test deck supplied with EPOCH, with a modified laser and longer runtime. Other classes of beam (Bessel etc) can be created similarly.
Injecting the laser at an angle
By setting up a phase shift as a function of space, it is possible to force wavefronts to arrive at different points on the boundary at different times (for 2D and 3D simulations). This allows the user to inject lasers with an angle to the boundary normal. Note that the laser boundary theory expects lasers to be injected perpendicular to the boundary, so it will almost always be better to obtain oblique incidence by rotating the target in EPOCH. Angular injection makes the most sense when multiple lasers are present with difference incident angles.
Angled incident laser profile
The setup for this is not entirely straightforward and requires a little bit of explanation. The above figure illustrates a laser being driven at an angle on the $x_{min}$ boundary. Different wave fronts cross the $y$-axis at different places and this forms a sinusoidal profile along $y$ that represents the phase. The wavelength of this profile is given by $\lambda_\phi = \lambda / \sin\theta$, where $\lambda$ is the wavelength of the laser and $\theta$ is the angle of the propagation direction with respect to the $x$-axis. The actual phase to use will be $\phi(y) = -k_\phi y = -2\pi y / \lambda_\phi$. It is negative because the phase of the wave is propagating in the positive $y$ direction. It is also necessary to alter the wavelength of the driver since this is given in the direction perpendicular to the boundary. The new wavelength to use will be $\lambda\cos\theta$. The figure shows the resulting $E_y$ field for a laser driven at an angle of $\pi / 6$. Note that since the boundary conditions in the code are derived for propagation perpendicular to the boundary, there will be artefacts on the scale of the grid for lasers driven at an angle.
The input deck used to generate this figure is given below:
begin:control
nx = 2000
ny = 2000
t_end = 50.e-15
x_min = -10.0e-6
x_max = 10.0e-6
y_min = -10.0e-6
y_max = 10.0e-6
end:control
begin:boundaries
bc_x_min = simple_laser
bc_x_max = simple_outflow
bc_y_min = simple_outflow
bc_y_max = simple_outflow
end:boundaries
begin:constant
laser_angle = pi / 6 # Set angle of laser w.r.t positive x unit vector
laser_fwhm = 2.0e-6 # Size of gaussian profile for laser electric field
laser_wavelength = 1.0e-6
laser_k = 2 * pi / laser_wavelength
end:constant
begin:laser
boundary = x_min
intensity_w_cm2 = 1.0e15
profile = gauss(y, 0, laser_fwhm / (2.0 * sqrt(loge(2.0))))
t_profile = 1
phase = -y * laser_k * sin(laser_angle) # Vary wave-front position in space
lambda = laser_wavelength * cos(laser_angle) # Space wave-fronts apart
# correctly along k
end:laser
begin:output
dt_snapshot = 10.e-15
grid = always
ey = always
end:output