Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.adass.org/adass/proceedings/adass94/jonesa.ps Äàòà èçìåíåíèÿ: Tue Jun 13 20:42:33 1995 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 00:48:31 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: arp 220

Astronomical Data Analysis Software and Systems IV
ASP Conference Series, Vol. 77, 1995
R. A. Shaw, H. E. Payne, and J. J. E. Hayes, eds.
Towards a General Definition for Spectroscopic Resolution
A. W. Jones, J. Bland­Hawthorn
Anglo­Australian Observatory, P.O. Box 296, Epping, NSW 2121
Australia
P. L. Shopbell
Dept. of Space Physics & Astronomy, Rice University, P.O. Box 1892,
Houston, TX 77251
Abstract. Judged by their instrumental profiles, spectrometers fall into
two basic classes---Lorentzian and Gaussian---with many other line profile
functions (sinc m functions, Voigt functions, Airy functions, etc.) falling
into one of these two categories in some limit. We demonstrate that
the Rayleigh, Sparrow, and Houston resolution criteria are of limited use
compared to the ``equivalent width'' of the line profile.
1. Introduction
Modern day spectrographs ultimately rely on the interference of a finite number
of beams that traverse different optical paths to form a signal (Bell 1972). The
spectrometer disperses the incoming light into a finite number of wavelength
(energy) intervals, where the size of the resolution element (ffi–) is set by the
bandwidth limit imposed by the dispersing element. Different dispersive tech­
niques produce a variety of instrumental profiles. A long­slit spectrometer in
the diffraction limit produces a sinc 2 wavelength response, a property shared
with acousto­optic filters. In practice, optical and mechanical defects within
either device tend to make the instrumental response more Gaussian in form.
The response of the Fourier Transform Spectrometer is fundamentally the sinc
function, although this response is commonly apodized to produce a profile with
better side­lobe behavior. An internally reflecting cavity (e.g., Fabry­Perot fil­
ter) generates an instrumental response given by the periodic Airy function. In
the limit of high finesse (the periodic interval \Delta– divided by the line FWHM
ffi –), the Airy function reduces to the Lorentzian function.
2. Resolution Criteria
The resolution element ffi–, or more formally the spectral purity, is the small­
est measurable wavelength difference at a given wavelength –. In the case of
rectangular and triangular functions, the (average) instrumental width is unam­
biguous; for more complex functions, a characteristic width can be more difficult
to define.
1

2
f(x) Area Sparrow
G exp(\Gamma ln 16x 2 =ffix 2 )
`
ffix
2
'r
ú
ln 2
ffi x
p
2 ln 2
L 1
1 + (2x=ffix) 2
`
ffi x
2
'
ú
ffi x
p
3
A 1
1 + ff sin 2 (úx=\Deltax)
\Deltax
p
1 + ff
lim
ff!1
ffi x
p
3
S sinc 2
` 2:7831x
ffi x
' `
ú
2:7831
'
ffi x 0:9364 ffi x
V 1
a
ú
+1 Z
\Gamma1
exp(\Gammay 2 )
(x
p
ln 16=ffix \Gamma y)
2
+ a 2
dy lim
a!0
`
ffi x
2
'r
ú
ln 2 lim
a!0
ffix
p
2 ln 2
V 2
a 2
p
ú
+1 Z
\Gamma1
exp(\Gammay 2 )
(2x=ffix \Gamma y) 2 + a 2
dy lim
a!1
`
ffi x
2
'p
ú
a
lim
a!1
ffi x
p
3
Table 1. Resolution criteria for common spectral line functions in
terms of the FWHM ffix (see Figure 1). The Airy function is periodic
over the interval \Deltax.
Rayleigh criterion. Lord Rayleigh (1879) first derived the resolved distance of
two identical, diffraction­limited point sources with the aid of Bessel functions.
This separation arises when the peak of one Bessel function falls on the first zero
point of the other function. The often quoted resolution criterion, 1:220f–=L,
where f is the focal ratio of the imaging system, was only intended for use in
this context. Thus, we do not investigate this criterion further.
Houston criterion. The usual metric in astronomy is to adopt the ``full width at
half maximum'' (FWHM) as a suitable definition of spectral purity ffi –. Houston
(1926) ventured that this property can be used to define the natural separation
of two identical lines which are resolved from each other.
Sparrow criterion. Sparrow (1916) suggested a clever alternative, which depends
on the property of the summed line profiles. As we move the lines closer together
from far apart, a minimum develops. Sparrow suggested that a natural definition
for resolution results at the line separation where the saddle point first develops
(i.e., the gradient at the peak of the summed profile is zero). More formally,
for an instrumental response given by f(x), then two sources are resolved at
a separation of oe L (Sparrow limit) when both of the following conditions are
satisfied:
d
dx
[f(x) + f(x + oe L )] = 0;
d 2
dx 2
[f(x) + f(x + oe L )] = 0:

3
Figure 1. (a) Five functions with the same FWHM that demonstrate
a gradual trend in the ratio of core power to wing power: (G) Gaus­
sian, (L) Lorentzian, (T) x \Gamma1 tanh x, (FS) Fraser­Suzuki function, (J)
the most extreme case to date discovered by AWJ. All of these func­
tions have continuous higher derivatives. In order for the function to
remain everywhere continuous, the neighborhood of the peak becomes
narrower (see inset) to compensate for the more extreme curvature near
the FWHM. (b) Three different possible criteria for the resolution of
two identical instrumental functions: (G) Gaussian, (L) Lorentzian,
(A) Airy, (S) sinc 2 , (V) Voigt.
3. Logical Dilemma
We illustrate a reductio ad absurdum arising from conventional definitions of
spectral resolution, with the aid of a series of functions that are everywhere
continuous and have the same FWHM, but have varying core to wing ratios (see
Figure 1(a)). Long­slit spectrometers usually have rather Gaussian (G) profiles,
whereas Fabry­Perot interferometers approach Lorentzian (L) at high finesse.
The extreme wing­to­core ratio function (J) is not unlike the spatial response
function of the Hubble Space Telescope prior to installation of the COSTAR op­
tics. The Houston criterion implies that all of the profiles have equal resolving
ability. In contrast, the Sparrow criterion would lead one to believe that nar­
row cores and large wings have better resolution capabilities (see Table 1 and
Figure 1(b)). This is clearly not physical. A resolved core does not guarantee
that two profiles are clearly separated, since the wing contribution remains un­
resolved. Intuitively, one way to see this is in terms of the shot noise constraint.
The uncertainty in finding the line centroid depends inversely on the signal un­

4
der the profile. As we increase the wing power, for a constant line flux, the
signal in the core decreases dramatically.
4. Comparison of Resolution Criteria
In Table 1, we compare five different functional forms which arise in obser­
vational astronomy. Each of the functions has been expressed in terms of its
FWHM ffi x; the Voigt function is expressed in two different ways to emphasize
its Lorentzian and Gaussian behavior in the different limits. The various cases
are illustrated in Figure 1(b). The figure shows that the Sparrow criterion is the
least stringent, followed by the Houston criterion. The equivalent width (area)
criterion is least forgiving to the Lorentzian (and therefore Airy) function. At
first glance, it would appear that the Houston criterion is sufficient to resolve two
lines. However, to avoid the logical dilemma described in the previous section,
we should say that the lines are properly resolved by the area criterion.
We propose that the equivalent width (the area of the line profile divided
by the peak height) is a better measure of the spectral purity of an instrumental
function. This is more physical for a number of reasons: (1) the equivalent
width, not the FWHM, constitutes the average width of the profile, (2) the
equivalent width is more representative of the total signal under the line profile,
(3) the equivalent width is a better discriminant of the wing behavior of a line
profile, and (4) the equivalent width has an entirely general definition for an
arbitrary positive function. For Gaussian spectrometers (e.g., long­slit devices),
the correction amounts to no more than 6%. For Lorentzian spectrometers (e.g.,
Fabry­Perot filters), this leads to a large correction factor (50--60%) to the more
standard use of the line profile FWHM.
Acknowledgments. AWJ acknowledges a studentship (austral winter, 1994)
at the Anglo­Australian Observatory.
References
Bell, R. J. 1972, Introductory Fourier Transform Spectroscopy (New York, Aca­
demic Press)
Fraser R. D. B., & Suzuki, E. 1969, Anal. Chem., 41, 37
Houston, W. V. 1926, ApJ, 64, 81
Rayleigh, Lord 1879, Phil. Mag., 8, 261
Sparrow, C. M. 1916, ApJ, 44, 76