Utilities for working with the Airy Pattern.

Functions
template<typename realT >
realT	mx::math::func::airyPattern (realT x)
	The classical Airy pattern.

template<typename realT >
realT	mx::math::func::airyPattern (realT x, realT eps)
	The centrally obscured Airy pattern.

template<typename realT >
realT	mx::math::func::airyPattern (realT x, realT y, realT A0, realT A, realT x0, realT y0, realT ps, realT eps)
	The general centrally obscured Airy pattern, with arbitrary center and platescale.

template<typename realT >
void	mx::math::func::airyPattern2D (realT *arr, size_t nx, size_t ny, const realT A0, const realT A, const realT x0, const realT y0, realT ps)
	Fill in an array with the 2D arbitrarily-centered classical Airy pattern.

template<typename realT >
realT	mx::math::func::seeingHalo (realT x, realT fwhm)
	Seeing Halo Profile.

template<typename realT >
realT	mx::math::func::airyPatternEnclosed (realT x)
	Calculate the fraction of enclosed power at a given radius for the unobscured Airy Pattern.

template<typename realT >
realT	mx::math::func::airyPatternEnclosed (realT x, realT eps)
	Calculate the fraction of enclosed power at a given radius for the centrally obscured Airy Pattern.

Function Documentation

◆ airyPattern() [1/3]

template<typename realT >

realT mx::math::func::airyPattern ( realT x )

The classical Airy pattern.

Returns the intensity distribution of the Airy pattern at a given \( \lambda/D \)

References: [1], [11], https://en.wikipedia.org/wiki/Airy_disk.

Template Parameters

realT is the floating point type used for arithmetic.

Parameters

[in] x is the separation in units of \( \lambda/D \).

Definition at line 57 of file airyPattern.hpp.

References mx::math::func::jinc(), and mx::math::six_fifths().

Referenced by mx::math::func::airyPattern(), mx::math::func::airyPattern2D(), and mx::AO::analysis::fourierPSDMap().

◆ airyPattern() [2/3]

template<typename realT >

realT mx::math::func::airyPattern	(	realT	x,
		realT	eps
	)

The centrally obscured Airy pattern.

Returns the intensity distribution of the centrally obscured Airy pattern at a given \( \lambda/D \)

References: [1], [11], https://en.wikipedia.org/wiki/Airy_disk.

Template Parameters

realT is the floating point type used for arithmetic.

Parameters

[in]	x	is the separation in units of \( \lambda/D \).
[in]	eps	is the ratio of the circular central obscuration diameter to the diameter.

Definition at line 73 of file airyPattern.hpp.

References mx::math::func::jinc(), and mx::math::six_fifths().

◆ airyPattern() [3/3]

template<typename realT >

realT mx::math::func::airyPattern	(	realT	x,
		realT	y,
		realT	A0,
		realT	A,
		realT	x0,
		realT	y0,
		realT	ps,
		realT	eps
	)

The general centrally obscured Airy pattern, with arbitrary center and platescale.

Returns the intensity distribution of the centrally obscured Airy pattern at a given \( \lambda/D \). This version allows for an arbitrary center, scaling, and platescale.

References: [1], [11], https://en.wikipedia.org/wiki/Airy_disk.

Template Parameters

realT is the floating point type used for arithmetic.

Parameters

[in]	x	is the x-coordinate in units of pixels
[in]	y	is the y-coordinate in units of pixels
[in]	A0	constant value added to the Airy pattern
[in]	A	peak scale of the Airy pattern.
[in]	x0	is the x-center in units of pixels
[in]	y0	is the y-center in units of pixels
[in]	ps	the platescale in \( (\lambda/D)/pixel \)
[in]	eps	is the ratio of the circular central obscuration diameter to the diameter.

Definition at line 93 of file airyPattern.hpp.

References mx::math::func::airyPattern(), and mx::math::six_fifths().

◆ airyPattern2D()

template<typename realT >

void mx::math::func::airyPattern2D	(	realT *	arr,
		size_t	nx,
		size_t	ny,
		const realT	A0,
		const realT	A,
		const realT	x0,
		const realT	y0,
		realT	ps
	)

Fill in an array with the 2D arbitrarily-centered classical Airy pattern.

At each pixel (x,y) of the array this computes:

\( f(x,y) = A_0 + A\times Airy(\sqrt(x^2+y^2) \)

Template Parameters

realT is the type to use for arithmetic

Parameters

[out]	arr	is the allocated array to fill in
[in]	nx	is the size of the x dimension of the array (rows)
[in]	ny	is the size of the y dimension of the array (columns)
[in]	A0	is the constant to add to the Gaussian
[in]	A	is the scaling factor (peak height = A-A0)
[in]	x0	is the x-coordinate of the center
[in]	y0	is the y-coordinate of the center
[in]	ps	the platescale in \( (\lambda/D)/pixel \)

Definition at line 119 of file airyPattern.hpp.

References mx::math::func::airyPattern(), and mx::math::six_fifths().

◆ airyPatternEnclosed() [1/2]

template<typename realT >

realT mx::math::func::airyPatternEnclosed ( realT x )

Calculate the fraction of enclosed power at a given radius for the unobscured Airy Pattern.

See Mahajan (1986) [11] and https://en.wikipedia.org/wiki/Airy_disk.

Returns: the fraction of enclosed power at radius x

Parameters

[in] x the radius

Definition at line 172 of file airyPattern.hpp.

References mx::math::six_fifths().

Referenced by mx::math::func::airyPatternEnclosed().

◆ airyPatternEnclosed() [2/2]

template<typename realT >

realT mx::math::func::airyPatternEnclosed	(	realT	x,
		realT	eps
	)

Calculate the fraction of enclosed power at a given radius for the centrally obscured Airy Pattern.

See Mahajan (1986) [11] and https://en.wikipedia.org/wiki/Airy_disk. If eps = 0, this calls the unobscured version.

Returns: the fraction of enclosed power at radius x

Parameters

[in]	x	the radius
[in]	eps	the central obscuration fraction

Definition at line 205 of file airyPattern.hpp.

References mx::math::func::airyPatternEnclosed(), and mx::math::six_fifths().

◆ seeingHalo()

template<typename realT >

realT mx::math::func::seeingHalo	(	realT	x,
		realT	fwhm
	)

Seeing Halo Profile.

A Moffat profile due to Roddier (1981)[20], see also Racine et al (1999)[19], which can be used to describe the seeing limited point-spread function of a telescope imaging through turbulence, or the halo in partially corrected imaging.

Returns: the value of the profile at x, in units of fractional flux per unit area.

Parameters

[in]	x	the separation in the same units as fwhm.
[in]	fwhm	the fwhm in arbitrary units. Note that this defines the area units of the density.

Definition at line 155 of file airyPattern.hpp.

References mx::math::six_fifths().

Functions

Function Documentation

◆ airyPattern() [1/3]

◆ airyPattern() [2/3]

◆ airyPattern() [3/3]

◆ airyPattern2D()

◆ airyPatternEnclosed() [1/2]

◆ airyPatternEnclosed() [2/2]

◆ seeingHalo()