[meanas.fdtd.misc] add basic pulse and beam shapes

meanas/fdtd/misc.py

@@ -0,0 +1,167 @@
+from typing import Callable
+from collections.abc import Sequence
+import logging
+
+import numpy
+from numpy.typing import NDArray, ArrayLike
+from numpy import pi
+
+
+logger = logging.getLogger(__name__)
+
+
+pulse_fn_t = Callable[[int | NDArray], tuple[float, float, float]]
+
+
+def gaussian_packet(
+        wl: float,
+        dwl: float,
+        dt: float,
+        turn_on: float = 1e-10,
+        one_sided: bool = False,
+        ) -> tuple[pulse_fn_t, float]:
+    """
+    Gaussian pulse (or gaussian ramp) for FDTD excitation
+
+    exp(-a*t*t) ==> exp(-omega * omega / (4 * a))   [fourier, ignoring leading const.]
+
+    FWHM_time is 2 * sqrt(2 * log(2)) * sqrt(2 / a)
+    FWHM_omega is 2 * sqrt(2 * log(2)) * sqrt(2 * a)  = 4 * sqrt(log(2) * a)
+
+    Args:
+        wl: wavelength
+        dwl: Gaussian's FWHM in wavelength space
+        dt: Timestep
+        turn_on: Max allowable amplitude at t=0
+        one_sided: If `True`, source amplitude never decreases after reaching max
+
+    Returns:
+        Source function: src(timestep) -> (envelope[tt], cos[... * tt], sin[... * tt])
+        Delay: number of initial timesteps for which envelope[tt] will be 0
+    """
+    logger.warning('meanas.fdtd.misc functions are still very WIP!')    # TODO
+    # dt * dw = 4 * ln(2)
+
+    omega = 2 * pi / wl
+    freq = 1 / wl
+    fwhm_omega = dwl * omega * omega / (2 * pi)         # dwl -> d_omega (approx)
+    alpha = (fwhm_omega * fwhm_omega) * numpy.log(2) / 8
+    delay = numpy.sqrt(-numpy.log(turn_on) / alpha)
+    delay = numpy.ceil(delay * freq) / freq     # force delay to integer number of periods to maintain phase
+    logger.info(f'src_time {2 * delay / dt}')
+
+    def source_phasor(ii: int | NDArray) -> tuple[float, float, float]:
+        t0 = ii * dt - delay
+        envelope = numpy.sqrt(numpy.sqrt(2 * alpha / pi)) * numpy.exp(-alpha * t0 * t0)
+
+        if one_sided and t0 > 0:
+            envelope = 1
+
+        cc = numpy.cos(omega * t0)
+        ss = numpy.sin(omega * t0)
+
+        return envelope, cc, ss
+
+    # nrm = numpy.exp(-omega * omega / alpha) / 2
+
+    return source_phasor, delay
+
+
+def ricker_pulse(
+        wl: float,
+        dt: float,
+        turn_on: float = 1e-10,
+        ) -> tuple[pulse_fn_t, float]:
+    """
+    Ricker wavelet (second derivative of a gaussian pulse)
+
+    t0 = ii * dt - delay
+    R = w_peak * t0 / 2
+    f(t) = (1 - 2 * (pi * f_peak * t0) ** 2) * exp(-(pi * f_peak * t0)**2
+         = (1 - (w_peak * t0)**2 / 2 exp(-(w_peak * t0 / 2) **2)
+         = (1 - 2 * R * R) * exp(-R * R)
+
+    # NOTE: don't use cosine/sine for J, just for phasor readout
+
+    Args:
+        wl: wavelength
+        dt: Timestep
+        turn_on: Max allowable amplitude at t=0
+
+    Returns:
+        Source function: src(timestep) -> (envelope[tt], cos[... * tt], sin[... * tt])
+        Delay: number of initial timesteps for which envelope[tt] will be 0
+    """
+    logger.warning('meanas.fdtd.misc functions are still very WIP!')    # TODO
+    omega = 2 * pi / wl
+    freq = 1 / wl
+    r0 = omega / 2
+
+    from scipy.optimize import root_scalar
+    delay_results = root_scalar(lambda xx: (1 - omega * omega * tt * tt / 2) * numpy.exp(-omega * omega / 4 * tt * tt) - turn_on, x0=0, x1=-2 / omega)
+    delay = delay_results.root
+    delay = numpy.ceil(delay * freq) / freq     # force delay to integer number of periods to maintain phase
+
+    def source_phasor(ii: int | NDArray) -> tuple[float, float, float]:
+        t0 = ii * dt - delay
+        rr = omega * t0 / 2
+        ff = (1 - 2 * rr * rr) * numpy.exp(-rr * rr)
+
+        cc = numpy.cos(omega * t0)
+        ss = numpy.sin(omega * t0)
+
+        return ff, cc, ss
+
+    return source_phasor, delay
+
+
+def gaussian_beam(
+        xyz: list[NDArray],
+        center: ArrayLike,
+        waist_radius: float,
+        wl: float,
+        tilt: float = 0,
+        ) -> NDArray[numpy.complex128]:
+    """
+    Gaussian beam
+    (solution to paraxial Helmholtz equation)
+
+    Default (no tilt) corresponds to a beam propagating in the -z direction.
+
+    Args:
+        xyz: List of [[x0, x1, ...], [y0, ...], [z0, ...]] positions specifying grid
+            locations at which the field will be sampled.
+        center: [x, y, z] location of beam waist
+        waist_radius: Beam radius at the waist
+        wl: Wavelength
+        tilt: Rotation around y axis. Default (0) has beam propagating in -z direction.
+    """
+    logger.warning('meanas.fdtd.misc functions are still very WIP!')    # TODO
+    w0 = waist_radius
+    grids = numpy.asarray(numpy.meshgrid(*xyz, indexing='ij'))
+    grids -= numpy.asarray(center)[:, None, None, None]
+
+    rot = numpy.array([
+        [ numpy.cos(tilt), 0, numpy.sin(tilt)],
+        [               0, 1,               0],
+        [-numpy.sin(tilt), 0, numpy.cos(tilt)],
+        ])
+
+    xx, yy, zz = numpy.einsum('ij,jxyz->ixyz', rot, grids)
+    r2 = xx * xx + yy * yy
+    z2 = zz * zz
+
+    zr = pi * w0 * w0 / wl
+    zr2 = zr * zr
+    wz2 = w0 * w0 * (1 + z2 / zr2)
+    wz = numpy.sqrt(wz2)        # == fwhm(z) / sqrt(2 * ln(2))
+
+    kk = 2 * pi / wl
+    Rz = zz * (1 + zr2 / z2)
+    gouy = numpy.arctan(zz / zr)
+
+    gaussian = w0 / wz * numpy.exp(-r2 / wz2) * numpy.exp(1j * (kk * zz + kk * r2 / 2 / Rz - gouy))
+
+    row = gaussian[:, :, gaussian.shape[2] // 2]
+    norm = numpy.sqrt((row * row.conj()).sum())
+    return gaussian / norm