beam2bl

This IDL facility computes a transfer (or window) function $b(\ell)$ for a circular beam profile $b(\theta)$.

Location in HEALPix directory tree: src/idl/misc/beam2bl.pro 

FORMAT

IDL> bl=beam2bl( beam, theta, lmax, [/ARCMIN , /DEGREES, /HELP, /RADIANS])

QUALIFIERS

beam

input beam profile $b(\theta)$

theta

angles $\theta$ (in arcmin, degrees or radians) at which the input beam $b(\theta)$ is defined

lmax

maximum multipole on which the output $b(\ell)$ is to be computed

KEYWORDS

/ARCMIN

if set, $\theta$ is in arcmin

/DEGREES

if set, $\theta$ is in degrees

/HELP

if set, prints out the help header and exits

/RADIANS

if set, $\theta$ is in radians

DESCRIPTION

Since the SH Transform of an arbitrary beam is

$$b_{\ell m} \int d{\textbf{r}}\ b({\textbf{r}})\ Y_{\ell m}^*({\textbf{r}})$$ = (1)

then, for a circular beam

$$b(\ell) b_{\ell 0} \sqrt{\frac{4 \pi}{2\ell+1}}$$ =

$$\int b(\theta) P_\ell(\theta) \sin(\theta)\ d\theta\ 2\pi$$ = (2)

where $P_\ell$ is the Legendre Polynomial, $b(\ell)$ is the beam window (or transfer) function returned by beam2bl and $b(\theta)$ is the beam radial profile expected as input of beam2bl.
IDL's routine INT_TABULATED is used to perform the integration.

RELATED ROUTINES

This section lists the routines related to beam2bl

idl

version 6.4 or more is necessary to run beam2bl.

bl2beam

facility to perform the inverse transform to beam2bl.

bl2fits

facility to write a $b(\ell)$ window function into a FITS file.

fits2cl

facility to read a $b(\ell)$ window function from a FITS file

EXAMPLE:

bl = gaussbeam(15.d0, 4000, 1)

theta = dindgen(4000)/100.

beam = bl2beam(bl, theta, /arcmin)

bl1 = beam2bl(beam, theta, 4000, /arcmin)

plot, bl1-bl

the example above generates a beam window function (defined for all $\ell$ in $\{0,\ldots,4000\}$) for a 15arcmin-FWHM gaussian beam, computes the beam profile for angles in [0,40] arcmin, computes back the beam window function from the beam profile and finally plots the difference between the beam window functions.