from typing import Union import numpy as np from astropy import units as u from .IPSF import IPSF from scipy.optimize import newton, fmin, bisect from scipy.special import j0, j1 from scipy.signal import fftconvolve from scipy.integrate import quad class Airy(IPSF): """ A class for modelling the PSF using an airy disk. """ @u.quantity_input(wl="length", d_aperture="length") def __init__(self, f_number: float, wl: u.Quantity, d_aperture: u.Quantity, osf: float, pixel_size: u.Quantity): """ Initialize a new PSF from a airy disk. Parameters ---------- f_number : float The working focal number of the optical system wl : Quantity The central wavelength which is used for calculating the PSF d_aperture : Quantity The diameter of the telescope's aperture. osf : float The oversampling factor to be used for oversampling the PSF with regards to the pixel size. pixel_size : Quantity The size of a pixel as length-quantity. """ self.__f_number = f_number self.__wl = wl self.__d_aperture = d_aperture self.__osf = osf self.__pixel_size = pixel_size def calcReducedObservationAngle(self, contained_energy: Union[str, int, float], jitter_sigma: u.Quantity = None, obstruction: float = 0.0) -> u.Quantity: """ Calculate the reduced observation angle in lambda / d_ap for the given contained energy. Parameters ---------- contained_energy : Union[str, int, float] The percentage of energy to be contained within a circle with the diameter reduced observation angle. jitter_sigma : Quantity Sigma of the telescope's jitter in arcsec obstruction : float The central obstruction as ratio A_ob / A_ap Returns ------- reduced_observation_angle: Quantity The reduced observation angle in lambda / d_ap """ # Calculate the reduced observation angle in lambda / D for the given encircled energy reduced_observation_angle = 0 * u.dimensionless_unscaled if type(contained_energy) == str: # Encircled energy is of type string if contained_energy.lower() == "peak": # For the peak value of the PSF, the observation angle becomes zero which leads to one exposed # pixel later in the code reduced_observation_angle = 0 * u.dimensionless_unscaled elif contained_energy.lower() == "fwhm": # Width of the FWHM of the airy disk reduced_observation_angle = 1.028 contained_energy = 0.4738 * u.dimensionless_unscaled if not np.isclose(obstruction, 0.0): # Use obstructed airy disk reduced_observation_angle = newton(lambda y: self.airy(np.pi * y, np.sqrt(obstruction)) - 0.5, reduced_observation_angle / 2) * 2 contained_energy = self.airy_int(np.pi * reduced_observation_angle / 2, np.sqrt(obstruction)) elif contained_energy.lower() == "min": # Width of the first minimum of the airy disk reduced_observation_angle = 1.22 * 2 contained_energy = 0.8377 * u.dimensionless_unscaled if not np.isclose(obstruction, 0.0): # Use obstructed airy disk reduced_observation_angle = fmin(lambda y: self.airy(np.pi * y, np.sqrt(obstruction)), reduced_observation_angle / 2, disp=False)[0] * 2 contained_energy = self.airy_int(np.pi * reduced_observation_angle / 2, np.sqrt(obstruction)) else: # Calculate the width numerically from the integral of the airy disk contained_energy = contained_energy / 100 * u.dimensionless_unscaled reduced_observation_angle = 2 * bisect( lambda y: self.airy_int(np.pi * y, np.sqrt(obstruction)) - contained_energy.value, 0, 100) if jitter_sigma is not None: # Convert jitter to reduced observation angle in lambda / d_ap jitter_sigma = jitter_sigma.to(u.rad).value * self.__d_aperture / self.__wl.to(u.m) # Calculate necessary grid length to accommodate the psf and 3-sigma of the gaussian grid_width = (reduced_observation_angle / 2 + 3 * jitter_sigma) # Calculate the reduced observation angle of a single detector pixel reduced_observation_angle_pixel = (self.__pixel_size / ( self.__f_number * self.__d_aperture) * self.__d_aperture / self.__wl) # Calculate the width of each grid element dx = reduced_observation_angle_pixel / self.__osf # Calculate the necessary number of points on the grid n_points = np.ceil(grid_width / dx).value # Calculate the corresponding x-coordinates of each grid element x = np.arange(1, n_points + 1) * dx # Calculate the psf from an airy disk for each element on the grid psf = self.airy(np.pi * x, np.sqrt(obstruction)) # Calculate the integral of the undisturbed airy disk in order to scale the result of the convolution total = np.sum(psf * x) * dx * 2 * np.pi # Mirror the PSF to the negative x-domain psf = np.concatenate((np.flip(psf), np.array([1]), psf)) # Calculate a gaussian kernel kernel = 1 / (2 * np.pi * jitter_sigma ** 2) * np.exp( - np.concatenate((np.flip(x), np.array([0]), x)) ** 2 / (2 * jitter_sigma ** 2)) # Normalize the kernel kernel = kernel / np.sum(kernel) # Convolve the PSF with gaussian kernel psf = fftconvolve(np.pad(psf, int(n_points), mode="constant", constant_values=0), kernel, mode="same") # Reduce the PSF to the positive x-domain psf = psf[int((psf.shape[0] - 1) / 2):] # Calculate the rolling integral of the PSF psf_int = np.cumsum(psf * np.arange(psf.shape[0])) * reduced_observation_angle_pixel.value / self.__osf # Scale the integral of the disturbed PSF equal to the undisturbed PSF psf_int = psf_int / psf_int[-1] * total / (4 / np.pi) # Calculate the reduced observation angle reduced_observation_angle = np.argmax( psf_int > contained_energy) * reduced_observation_angle_pixel.value / self.__osf * 2 return reduced_observation_angle * u.dimensionless_unscaled @staticmethod def airy(x: Union[float, np.ndarray], obstruction: float = None): """ Calculate function values of the airy disk Parameters ---------- x : Union[float, np.ndarray] radial coordinate to calculate the function value for. obstruction : float The linear central obstruction ratio of the aperture. Returns ------- res : Union[float, np.ndarray] The function values of the airy disk at the given coordinates """ # Standardize input values if not isinstance(x, np.ndarray): x = np.array([x]) # Initialize return values and assign values for the singularity at x=0 res = np.zeros(len(x)) res[np.isclose(x, 0.0)] = 1.0 x_temp = x[np.invert(np.isclose(x, 0.0))] if obstruction and not np.isclose(obstruction, 0.0): # Use obstructed airy disk # See also https://en.wikipedia.org/wiki/Airy_disk#Obscured_Airy_pattern res[np.invert(np.isclose(x, 0.0))] = 1 / (1 - obstruction ** 2) ** 2 * ( 2 * (j1(x_temp) - obstruction * j1(obstruction * x_temp)) / x_temp) ** 2 else: # Use unobstructed airy disk # See also https://en.wikipedia.org/wiki/Airy_disk#Mathematical_formulation res[np.invert(np.isclose(x, 0.0))] = (2 * j1(x_temp) / x_temp) ** 2 # Unbox arrays of length 1 if len(res) == 1: res = res[0] return res @staticmethod def airy_int(x: float, obstruction: float = None): """ Calculate the integral of the airy disk from 0 to x. Parameters ---------- x : float The upper limit for the integration. obstruction : float The linear central obstruction ratio of the aperture. Returns ------- res : float The integral of the airy disk. """ if np.isclose(x, 0.0): # Short circuit for an integration-range of length 0 return 0. else: if obstruction and not np.isclose(obstruction, 0.0): # Use integral of obstructed airy disk # See also https://en.wikipedia.org/wiki/Airy_disk#Obscured_Airy_pattern return 1 / (1 - obstruction ** 2) * (1 - j0(x) ** 2 - j1(x) ** 2 + obstruction ** 2 * ( 1 - j0(obstruction * x) ** 2 - j1(obstruction * x) ** 2) - 4 * obstruction * quad( lambda y: j1(y) * j1(obstruction * y) / y, 0, x, limit=100, epsrel=1e-6)[0]) else: # Use unobstructed airy disk # See also https://en.wikipedia.org/wiki/Airy_disk#Mathematical_formulation return 1 - j0(x) ** 2 - j1(x) ** 2 def mapToGrid(self, grid: np.ndarray) -> np.ndarray: """ Map the integrated PSF values to a sensor grid. Parameters ---------- grid : ndarray The grid to map the values to. The values will only be mapped onto entries with the value 1. Returns ------- grid : ndarray The grid with the mapped values. """ pass