phase.py

"""
Repository of common analytic phase patterns.
"""

import numpy as np
from scipy import special
from math import factorial

from slmsuite.misc.math import REAL_TYPES
from slmsuite.holography.toolbox import _process_grid

def blaze(grid, vector=(0, 0), offset=0):
    r"""
    Returns a simple `blaze <https://en.wikipedia.org/wiki/Blazed_grating>`_,
    a linear phase ramp, toward a given vector in :math:`k`-space.

    .. math:: \phi(\vec{x}) = 2\pi \cdot \vec{k}_{norm} \cdot \vec{x}_{norm} + o

    Parameters
    ----------
    grid : (array_like, array_like) OR :class:`~slmsuite.hardware.slms.slm.SLM`
        :math:`\vec{x}_{norm}`. Meshgrids of normalized :math:`\frac{x}{\lambda}` coordinates
        corresponding to SLM pixels, in ``(x_grid, y_grid)`` form.
        These are precalculated and stored in any :class:`~slmsuite.hardware.slms.slm.SLM`, so
        such a class can be passed instead of the grids directly.
    vector : (float, float)
        :math:`\vec{k}_{norm}`. Blaze vector in normalized :math:`\frac{k_x}{k}` units.
        See :meth:`~slmsuite.holography.toolbox.convert_blaze_vector()`
    offset :
        Phase offset for this blaze.

    Returns
    -------
    numpy.ndarray
        The phase for this function.
    """
    (x_grid, y_grid) = _process_grid(grid)

    # Optimize phase construction based on context.
    if vector[0] == 0 and vector[1] == 0:
        return np.zeros_like(x_grid) + offset
    elif vector[0] == 0:
        return 2 * np.pi * (vector[1] * y_grid) + offset
    elif vector[1] == 0:
        return 2 * np.pi * (vector[0] * x_grid) + offset
    else:
        return 2 * np.pi * (vector[0] * x_grid + vector[1] * y_grid) + offset


def lens(grid, f=(np.inf, np.inf)):
    r"""
    Returns a simple
    `thin parabolic lens <https://en.wikipedia.org/wiki/Thin_lens#Physical_optics>`_.
    When ``f`` is isotropic,

    .. math:: \phi(\vec{x}) = \frac{\pi}{f}|\vec{x}|^2

    Otherwise,

    .. math:: \phi(x, y) = \pi \left[\frac{x^2}{f_x} + \frac{y^2}{f_y} \right]

    Parameters
    ----------
    grid : (array_like, array_like) OR :class:`~slmsuite.hardware.slms.slm.SLM`
        Meshgrids of normalized :math:`\frac{x}{\lambda}` coordinates
        corresponding to SLM pixels, in ``(x_grid, y_grid)`` form.
        These are precalculated and stored in any :class:`~slmsuite.hardware.slms.slm.SLM`, so
        such a class can be passed instead of the grids directly.
    f : float OR (float, float)
        Focus in normalized :math:`\frac{x}{\lambda}` units.
        Scalars are interpreted as a non-cylindrical isotropic lens.
        Future: add a ``convert_focal_length`` method to parallel
        :meth:`.convert_blaze_vector()`
        Defaults to infinity (no lens).

    Returns
    -------
    numpy.ndarray
        The phase for this function.
    """
    (x_grid, y_grid) = _process_grid(grid)

    # Parse focal length.
    if isinstance(f, REAL_TYPES):
        f = [f, f]
    if isinstance(f, (list, tuple, np.ndarray)):
        f = np.squeeze(f)

        assert f.shape == (2,)
        assert not np.any(f == 0), "Cannot interpret a focal length of zero."

    # Optimize phase construction based on context (for speed, to avoid square, etc).
    if np.isfinite(f[0]) and np.isfinite(f[1]):
        return (np.pi / f[0]) * np.square(x_grid) + (np.pi / f[1]) * np.square(y_grid)
    elif np.isfinite(f[0]) and np.isfinite(f[1]):
        return (np.pi / f[0]) * np.square(x_grid)
    elif np.isfinite(f[1]):
        return (np.pi / f[1]) * np.square(y_grid)
    else:
        return np.zeros_like(x_grid)


def axicon(grid, f=(np.inf, np.inf), w=None):
    r"""
    Returns an `axicon <https://en.wikipedia.org/wiki/Axicon>`_ lens, the phase farfield for a Bessel beam.
    A (elliptically)-cylindrical axicon blazes according to :math:`\vec{k}_g = w / \vec{f} / 2` where
    :math:`w` is the radius of the axicon. With a flat input amplitude over
    :math:`[-w, w]`, this will produce a Bessel beam centered at :math:`z = \vec{f}`.

    .. math:: \phi(\vec{x}) = 2\pi \cdot \vec{k}_g \cdot |\vec{x}|

    Parameters
    ----------
    grid : (array_like, array_like) OR :class:`~slmsuite.hardware.slms.slm.SLM`
        Meshgrids of normalized :math:`\frac{x}{\lambda}` coordinates
        corresponding to SLM pixels, in ``(x_grid, y_grid)`` form.
        These are precalculated and stored in any :class:`~slmsuite.hardware.slms.slm.SLM`, so
        such a class can be passed instead of the grids directly.
    f : float OR (float, float)
        Focal length (center of the axicon diamond) in normalized :math:`\frac{x}{\lambda}` units.
        Scalars are interpreted as a non-cylindrical isotropic axicon.
        Defaults to infinity (no axicon).
    w : float OR None
        See :meth:`~slmsuite.holography.toolbox._determine_source_radius()`.

    Returns
    -------
    numpy.ndarray
        The phase for this function.
    """
    (x_grid, y_grid) = _process_grid(grid)

    w = _determine_source_radius(grid, w)

    if isinstance(f, REAL_TYPES):
        f = [f, f]
    if isinstance(f, (list, tuple, np.ndarray)):
        f = np.squeeze(f)

        assert f.shape == (2,)
        assert not np.any(f == 0), "Cannot interpret a focal length of zero."

    angle = [w / f[0] / 2, w / f[1] / 2]    # Notice that this fraction is in radians.

    # Optimize phase construction based on context (for speed, to avoid sqrt, etc).
    if angle[0] == 0 and angle[1] == 0:
        return np.zeros_like(x_grid)
    elif angle[0] == 0:
        return 2 * np.pi * np.abs(y_grid) * angle[1]
    elif angle[1] == 0:
        return 2 * np.pi * np.abs(x_grid) * angle[0]
    else:
        return 2 * np.pi * np.sqrt(np.square(x_grid * angle[0]) + np.square(y_grid * angle[1]))


def zernike(grid, n, m, aperture=None):
    r"""
    Returns a single real `Zernike polynomial <https://en.wikipedia.org/wiki/Zernike_polynomials>`_.

    Parameters
    ----------
    grid : (array_like, array_like) OR :class:`~slmsuite.hardware.slms.slm.SLM`
        Meshgrids of normalized :math:`\frac{x}{\lambda}` coordinates
        corresponding to SLM pixels, in ``(x_grid, y_grid)`` form.
        These are precalculated and stored in any :class:`~slmsuite.hardware.slms.slm.SLM`, so
        such a class can be passed instead of the grids directly.
    n, m : int
        Cartesian Zernike index defining the polynomial.
    aperture : {"circular", "elliptical", "cropped"} OR (float, float) OR None
        See :meth:`.zernike_sum()`.

    Returns
    -------
    numpy.ndarray
        The phase for this function.
    """
    return zernike_sum(grid, (((n, m), 1), ), aperture=aperture)


def zernike_sum(grid, weights, aperture=None):
    r"""
    Returns a summation of
    `Zernike polynomial <https://en.wikipedia.org/wiki/Zernike_polynomials>`_
    in a computationally-efficient manner. To improve performance, especially for higher
    order polynomials, we store a cache of Zernike coefficients to avoid regeneration.
    See the below example to generate :math:`Z_{20} - Z_{21} + Z_{31}`.

    .. highlight:: python
    .. code-block:: python

        zernike_sum_phase = toolbox.phase.zernike_sum(
            grid=slm,
            weights=(   ((2, 0),  1),       # Z_20
                        ((2, 1), -1),       # Z_21
                        ((3, 1),  1)    ),  # Z_31
            aperture="circular"
        )


    Note
    ~~~~
    There are different schemes to index Zernike polynomials.
    We use the indexing defined in `this paper <https://doi.org/10.1117/12.294412>`_,
    along with the algorithm defined there.
    Other packages use different schemes, sometimes defining
    :math:`m' = l = n - 2m`. Take care to avoid confusion.

    Important
    ~~~~~~~~~
    Zernike polynomials are canonically defined on a circular aperture. However, we may
    want to use these polynomials on other apertures (e.g. a rectangular SLM).
    Cropping this aperture breaks the orthogonality and normalization of the set, but
    this is fine for many applications. While it is possible to orthonormalize the
    cropped set, we do not do so in :mod:`slmsuite`, as this is not critical for target
    applications such as aberration correction.

    Parameters
    ----------
    grid : (array_like, array_like) OR :class:`~slmsuite.hardware.slms.slm.SLM`
        Meshgrids of normalized :math:`\frac{x}{\lambda}` coordinates
        corresponding to SLM pixels, in ``(x_grid, y_grid)`` form.
        These are precalculated and stored in any :class:`~slmsuite.hardware.slms.slm.SLM`, so
        such a class can be passed instead of the grids directly.
    weights : list of ((int, int), float)
        Which Zernike polynomials to sum. The ``(int, int)`` is the index ``(n, m)``, 
        which correspond to the azimuthal degree and order of the polynomial.
        The ``float`` is the weight for the given index.
    aperture : {"circular", "elliptical", "cropped"} OR (float, float) OR None
        How to scale the polynomials relative to the grid shape. This is relative
        to the :math:`R = 1` edge of a standard Zernike pupil.

        ``"circular"``, ``None``
          The circle is scaled isotropically until the pupil edge touches the grid edge.
          This is the default aperture.
        ``"elliptical"``
          The circle is scaled anisotropically until each cartesian pupil edge touches a grid
          edge. Generally produces and ellipse.
        ``"cropped"``
          The circle is scaled isotropically until the rectangle of the grid is
          circumscribed by the circle.
        ``(float, float)``
          Custom scaling. These values are multiplied to the ``x_grid`` and ``y_grid``
          directly, respectively. The edge of the pupil corresponds to where
          ``x_grid**2 + y_grid**2 = 1``.

    Returns
    -------
    numpy.ndarray
        The phase for this function.
    """
    # Parse passed values
    (x_grid, y_grid) = _process_grid(grid)

    if aperture is None:
        aperture = "circular"

    if isinstance(aperture, str):
        if aperture == "elliptical":
            x_scale = 1 / np.nanmax(x_grid)
            y_scale = 1 / np.nanmax(y_grid)
        elif aperture == "circular":
            x_scale = y_scale = 1 / np.amin([np.nanmax(x_grid), np.nanmax(y_grid)])
        elif aperture == "cropped":
            x_scale = y_scale = 1 / np.sqrt(np.nanmax(np.square(x_grid) + np.square(y_grid)))
        else:
            raise ValueError("NotImplemented")
    elif isinstance(aperture, (list, tuple)) and len(aperture) == 2:
        x_scale = aperture[0]
        y_scale = aperture[1]
    else:
        raise ValueError("Type {} not recognized.".format(type(aperture)))

    # At the end, we're going to set the values outside the aperture to zero.
    # Make a mask for this if it's necessary.
    mask = np.square(x_grid * x_scale) + np.square(y_grid * y_scale) <= 1
    use_mask = np.any(mask == 0)

    if use_mask:
        x_grid_scaled = x_grid[mask] * x_scale
        y_grid_scaled = y_grid[mask] * y_scale
    else:
        x_grid_scaled = x_grid * x_scale
        y_grid_scaled = y_grid * y_scale

    # Now find the coefficients for polynomial terms x^ay^b. We want to only compute
    # x^ay^b once because this is an operation on a large array. In contrast, summing
    # the coefficients of the same terms is simple and fast scalar operations.
    summed_coefficients = {}

    for (key, weight) in weights:
        coefficients = _zernike_coefficients(key[0], key[1])

        for power_key, factor in coefficients.items():
            power_factor = factor * weight
            if power_key in summed_coefficients:
                summed_coefficients[power_key] += power_factor
            else:
                summed_coefficients[power_key] = power_factor

    # Finally, build the polynomial.
    canvas = np.zeros(x_grid.shape)

    for power_key, factor in summed_coefficients.items():
        if factor != 0:
            if power_key == (0,0):
                if use_mask:
                    canvas[mask] += factor
                else:
                    canvas += factor
            else:
                if use_mask:
                    canvas[mask] += factor * np.power(x_grid_scaled, power_key[0]) * np.power(y_grid_scaled, power_key[1])
                else:
                    canvas += factor * np.power(x_grid_scaled, power_key[0]) * np.power(y_grid_scaled, power_key[1])

    return canvas

_zernike_cache = {}

def _zernike_coefficients(n, m):
    """
    Returns the coefficients for the :math:`x^ay^b` terms of the real cartesian Zernike polynomial
    of index `(`n, m)``. This is returned as a dictionary of form ``{(a,b) : coefficient}``.
    Uses the algorithm and indexing given in `this paper <https://doi.org/10.1117/12.294412>`_.
    """
    n = int(n)
    m = int(m)

    assert 0 <= m <= n, "Invalid cartesian Zernike index."

    # Generate coefficients only if we have not already generated.
    key = (n, m)

    if not key in _zernike_cache:
        zernike_this = {}

        # Define helper variables.
        l = n - 2 * m

        if l % 2:   # If even
            q = int((abs(l) - 1) / 2)
        else:
            if l > 0:
                q = int(abs(l)/2 - 1)
            else:
                q = int(abs(l)/2)

        if l <= 0:
            p = 0
        else:
            p = 1

        l = abs(l)
        m = int((n-l)/2)

        # Helper function
        def comb(n, k):
            return factorial(n) / (factorial(k) * factorial(n-k))

        # Finding the coefficients is a summed combinatorial search.
        # This is why we cache: so we don't have to do this many times,
        # especially for higher order polynomials and the corresponding cubic scaling.
        for i in range(q+1):
            for j in range(m+1):
                for k in range(m-j+1):
                    factor = -1 if (i + j) % 2 else 1
                    factor *= comb(l, 2 * i + p)
                    factor *= comb(m - j, k)
                    factor *= (float(factorial(n - j))
                        / (factorial(j) * factorial(m - j) * factorial(n - m - j)))

                    power_key = (n - 2*(i + j + k) - p, 2 * (i + k) + p)

                    # Add this coefficient to the element in the dictionary
                    # corresponding to the right power.
                    if power_key in zernike_this:
                        zernike_this[power_key] += factor
                    else:
                        zernike_this[power_key] = factor

        # Update the cache. Remove all factors that have cancelled out (== 0).
        _zernike_cache[key] = {power_key: factor for power_key, factor in zernike_this.items() if factor != 0}

    return _zernike_cache[key]

# Structured light

def _determine_source_radius(grid, w=None):
    r"""
    Helper function to determine the assumed Gaussian source radius for various
    structured light conversion functions. This is important because structured light
    conversions need knowledge of the size of the incident Gaussian beam.
    For example, see the ``w`` parameter in
    :meth:`~slmsuite.holography.toolbox.phase.laguerre_gaussian()`.

    Note
    ~~~~
    Future work: when ``grid`` is a :class:`~slmsuite.hardware.slms.slm.SLM` which has completed
    :meth:`~slmsuite.hardware.cameraslm.FourierSLM.fourier_calibration()`, this function should fit
    (and cache?) :attr:`~slmsuite.hardware.slms.slm.amplitude_measured` to a Gaussian
    and use the resulting width (and center?).

    Parameters
    ----------
    grid : (array_like, array_like) OR :class:`~slmsuite.hardware.slms.slm.SLM`
        Meshgrids of normalized :math:`\frac{x}{\lambda}` coordinates
        corresponding to SLM pixels, in ``(x_grid, y_grid)`` form.
        These are precalculated and stored in any :class:`~slmsuite.hardware.slms.slm.SLM`, so
        such a class can be passed instead of the grids directly.
    w : float OR None
        The radius of the phase pattern in normalized :math:`\frac{x}{\lambda}` units.
        To produce perfect structured beams, this radius is equal to the radius of
        the gaussian profile of the source (ideally not clipped by the SLM).
        If ``w`` is left as ``None``, ``w`` is set to a quarter of the smallest normalized screen dimension.

    Returns
    -------
    float
        Determined radius. In normalized units.
    """
    (x_grid, y_grid) = _process_grid(grid)

    if w is None:
        return np.min([np.amax(x_grid), np.amax(y_grid)]) / 4
    else:
        return w


def laguerre_gaussian(grid, l, p, w=None):
    r"""
    Returns the phase farfield for a
    `Laguerre-Gaussian <https://en.wikipedia.org/wiki/Gaussian_beam#Laguerre-Gaussian_modes>`_
    beam.

    This function is especially useful to hone and validate SLM alignment. Perfect alignment will
    result in concentric and uniform fringes for higher order beams. Focusing issues, aberration,
    or pointing misalignment will mitigate this.

    Parameters
    ----------
    grid : (array_like, array_like) OR :class:`~slmsuite.hardware.slms.slm.SLM`
        Meshgrids of normalized :math:`\frac{x}{\lambda}` coordinates
        corresponding to SLM pixels, in ``(x_grid, y_grid)`` form.
        These are precalculated and stored in any :class:`~slmsuite.hardware.slms.slm.SLM`, so
        such a class can be passed instead of the grids directly.
    l : int
        The azimuthal wavenumber, or orbital angular momentum. Can be negative.
    p : int
        The radial wavenumber. Should be non-negative.
    w : float OR None
        See :meth:`~slmsuite.holography.toolbox._determine_source_radius()`.

    Returns
    -------
    numpy.ndarray
        The phase for this function.
    """
    (x_grid, y_grid) = _process_grid(grid)

    w = _determine_source_radius(grid, w)

    theta_grid = np.arctan2(x_grid, y_grid)
    radius_grid = y_grid * y_grid + x_grid * x_grid

    return np.mod(
        l * theta_grid
        + np.pi
        * np.heaviside(-special.genlaguerre(p, np.abs(l))(2 * radius_grid / w / w), 0)
        + np.pi,
        2 * np.pi,
    )


def hermite_gaussian(grid, n, m, w=None):
    r"""
    Returns the phase farfield for a
    `Hermite-Gaussian <https://en.wikipedia.org/wiki/Gaussian_beam#Hermite-Gaussian_modes>`_
    beam. Uses the formalism described by `this paper <https://doi.org/10.1364/AO.54.008444>`_.

    Parameters
    ----------
    grid : (array_like, array_like) OR :class:`~slmsuite.hardware.slms.slm.SLM`
        Meshgrids of normalized :math:`\frac{x}{\lambda}` coordinates
        corresponding to SLM pixels, in ``(x_grid, y_grid)`` form.
        These are precalculated and stored in any :class:`~slmsuite.hardware.slms.slm.SLM`, so
        such a class can be passed instead of the grids directly.
    n, m : int
        The horizontal (``n``) and vertical (``m``) wavenumbers. ``n = m = 0`` yields a flat
        phase or a standard Gaussian beam.
    w : float
        See :meth:`~slmsuite.holography.toolbox._determine_source_radius()`.

    Returns
    -------
    numpy.ndarray
        The phase for this function.
    """
    (x_grid, y_grid) = _process_grid(grid)
    w = _determine_source_radius(grid, w)

    factor = np.sqrt(2) / w

    # Generate the amplitude of a Hermite-Gaussian mode.
    phase = special.hermite(n)(factor * x_grid) * special.hermite(m)(factor * y_grid)

    # This is real, so the phase is just the sign of the mode. This produces a
    # checkerboard pattern. Probably could make this faster by bitflipping rows and columns.
    phase[phase < 0] = 0
    phase[phase > 0] = np.pi

    return phase


def ince_gaussian(grid, p, m, parity=1, ellipticity=1, w=None):
    r"""
    **(NotImplemented)** Returns the phase farfield for an
    `Ince-Gaussian <https://en.wikipedia.org/wiki/Gaussian_beam#Ince-Gaussian_modes>`_
    beam.
    `Consider <https://doi.org/10.1364/OL.29.000144>`_
    `using <https://doi.org/10.1364/AO.54.008444>`_
    `these <https://doi.org/10.3390/jimaging8050144>`_
    `references <https://en.wikipedia.org/wiki/Elliptic_coordinate_system>`_.

    Parameters
    ----------
    grid : (array_like, array_like) OR :class:`~slmsuite.hardware.slms.slm.SLM`
        Meshgrids of normalized :math:`\frac{x}{\lambda}` coordinates
        corresponding to SLM pixels, in ``(x_grid, y_grid)`` form.
        These are precalculated and stored in any :class:`~slmsuite.hardware.slms.slm.SLM`, so
        such a class can be passed instead of the grids directly.
    p : int
        Ince polynomial order.
    m : int
        Ince polynomial degree.
    parity : {1, -1, 0}
        Whether to produce an even (1), odd (-1), or helical (0) Ince polynomial. A helical
        polynomial is the linear combination of even and odd polynomials.

        .. math:: IG^h_{p,m} = IG^e_{p,m} + iIG^o_{p,m}

    ellipticity : float
        Ellipticity of the beam. The semifocal distance is equal to ``ellipticity * w``,
        where the foci are the points which define the elliptical coordinate system.
    w : float
        See :meth:`~slmsuite.holography.toolbox._determine_source_radius()`.

    Returns
    -------
    numpy.ndarray
        The phase for this function.
    """
    (x_grid, y_grid) = _process_grid(grid)
    w = _determine_source_radius(grid, w)

    if parity == 1:
        assert 0 <= m <= p
    else:
        assert 1 <= m <= p

    complex_grid = x_grid + 1j * y_grid

    factor = 1 / (w * np.sqrt(ellipticity / 2))

    elliptic_grid = np.arccosh(complex_grid * factor)

    raise NotImplementedError()


def matheui_gaussian(grid, r, q, w=None):
    """
    **(NotImplemented)** Returns the phase farfield for a
    `Matheui-Gaussian <https://doi.org/10.1364/AO.49.006903>`_ beam.

    Returns
    -------
    numpy.ndarray
        The phase for this function.
    """
    (x_grid, y_grid) = _process_grid(grid)
    w = _determine_source_radius(grid, w)

    raise NotImplementedError()