Photometry with FabryPerot Spectrometers
J. BlandHawthorn
Abstract
At optical and infrared wavelengths, imaging FabryPerot devices are used
in three different ways: (i) to obtain a single spectrum of a diffuse source
which fills a large fraction of the aperture, (ii) to obtain a monochromatic
image within a field defined by the Jacquinot spot, and (iii) to obtain
a spectrum at each pixel position over a wide field by tuning the etalon.
We review the theoretical basis for FabryPerot photometry and summarize
the calibration procedures for the different applications. We discuss methods
for recognizing and dealing with artefacts (scattered light, atmospheric
effects, etc.) which can seriously comprise the photometric integrity of
the data if left untreated.
1. Introduction
Imaging FabryPerot interferometers are now in common use at several major
observatories and operate at both optical and infrared wavelengths. Traditionally,
FabryPerots are employed to perform studies of extended gaseous nebulae.
Examples include outflow sources (starburst and active galaxies, HerbigHaro
systems) and normal disk galaxies. Some groups have utilised the angular
spectral coverage to detect diffuse sources (HI/Lya
clouds in optical emission). More recently, scanning FabryPerots have
been used to construct spectral line profiles at many pixel positions across
a large format CCD. In many instances, these spectra are used simply to
obtain kinematic information (line widths, radial velocities) in a strong
emission line. It is less common to see these instruments employed as spectrophotometers
where the observed signal is calibrated to exoatmospheric flux units. This
may be due, in part, to a perception that the nature of the Airy function
makes FabryPerots unreliable photometers.
In this review, we describe reliable methods for fluxcalibrating FabryPerot
data with the aid of worked examples. We start by comparing the Airy function
to other well known functional forms in order to emphasize its distinct
properties. This leads to the concept of the `effective photometric bandpass'
which has important implications for the photometric calibration procedure.
Accurate photometry does require that the monochromatic and whitelight
response of the system are well understood, with the aid of simple numerical
simulations if need be. There are many potential pitfalls in the analysis:
we illustrate a few of these with data from instruments in Australia, Chile
and Hawaii.
Figure 1: A
FabryPerot etalon comprising two glass plates with highly reflective interior
surfaces, \cal R_{1} and \cal R_{2}, and antireflective
exterior coatings, \cal AR_{1} and \cal AR_{2}.
2. Airy function
The most direct route to the Airy function, the instrumental response of
the FabryPerot, is to use complex exponential notation. In Fig. 1, an
incoming plane wave with wavelength l at an
angle q to the optical axis enters the etalon
cavity and performs a series of internal reflections. If the highly reflective
inner surfaces have reflectivities of \cal R_{1} and \cal R_{2},
we can sum over the complex amplitudes of the outgoing plane waves such
that
\cal I = 1 + \cal R_{1}\cal R_{2}
e^{2 i d} + \cal R_{1}^{2}
\cal R_{2}^{2} e^{4 i d}
+ .... = 
1
1\cal R_{1} \cal R_{2} e^{2 i d} 


(1) 
in which 2d is the phase difference between
successive rays. The total transmitted intensity is proportional to the
squared modulus of the complex amplitude or
\cal A = \cal I\cal I^{*} = 
1
1+ 
4\cal N^{2
}p^{2} 
sin^{2} 2pml
l^{1}cosq 



(2) 
where the refractive index and the plate separation of the cavity are m
and l respectively, and \cal N = pÖ[4]\cal
R_{1}\cal R_{2}(1Ö{\cal
R_{1}\cal R_{2}})^{1} . Clearly, the Airy function
has a series of periodic maxima whenever
which is the well known equation of constructive interference in the mth
order.
The quantity \cal N is called the reflective finesse and depends only
on the values of \cal R_{1} and \cal R_{2}. It is normal
procedure to manufacture an etalon with two identical coatings such that
\cal R_{1} = \cal R_{2}. However, in Section 8, we illustrate
an important manifestation of a resonating cavity with very different reflection
coefficients.
Figure 2: Three
cyclic functions listed in Table 1 (column 2) shown at low finesse to emphasize
their differences.
Figure 3: The
integral of the Airy curve as a function of finesse normalized with respect
to the Gaussian and Lorentzian integrals. The asymptotic limit of the upper
curve is Ö{pln2}.




f(x) 
òf(x) dx 






G 
exp[ln16 mod[x,Dx]^{2} / (dx)^{2}] 
([(dx)/2]) Ö{[(p)/ln2]}
erf[([(2Ö{ln2})/(dx)])
mod[x,Dx]] 



L 
(1+([2/(dx)])^{2} mod[x,Dx]^{2})^{1} 
([(dx)/2]) tan^{1} [([2/(dx)])
mod[x,Dx]] 



A 
(1+ asin^{2}([(px)/(Dx)]))^{1} 
([(Dx)/(pÖ{1+a})])
tan^{1} [ Ö{1+a}
tan([(px)/(Dx)])] 



Table 1: Cyclic
functions which are periodic over Dx with FWHM
dx. The (G)aussian, (L)orentzian and (A)iry
functions are illustrated in Fig. 2. The mod function is the modulo
function and a = ([(2 Dx)/(pdx)])^{2}.
Note that for large a, ([(Dx)/(pÖ{1+a})])
»
([(dx)/2]). The gap scanning variable x is offset
by [(Dx)/2] in practice.
In order to arrive at the correct calibration procedure, it is important
to understand the nature of the Airy function. Fig. 2 illustrates the three
cyclic functions in Table 1 and shows that the area of the Airy function
always exceeds the integral of the other functions for a given spectroscopic
resolution. By analogy with the Lorentzian profile, the coefficient of
the sin^{2} factor in equation (2) determines the width of the
function. The quantity Dx/dx
is called the effective finesse and Dx
is the periodic free spectral range. In Fig. 3, we illustrate how the Airy
function, when normalized to the Gaussian and Lorentzian functions, depends
on \cal N. In practice, there are factors other than coating reflectivities
which contribute to the effective finesse  aperture effects, imperfections
within the plate coatings, etc.  some of which can serve to make the instrumental
response more Gaussian than Lorentzian in form (Atherton et al 1981).
Beyond a finesse of roughly 40, the Airy function is highly Lorentzian.
The reason for this is clear when looking at how the Airy profile has been
written in Table 1. At high finesse (Dx >>
dx), if we expand x about the peak of the profile, the small angle
formula reduces the Airy function to the Lorentzian form. The normalized
Airy integral depends only weakly on finesse at large finesse values.
3. Jacquinot advantage
There are several approaches to deriving the Jacquinot advantage (Roesler
1974; Thorne 1988), i.e. the throughput advantage of the FabryPerot interferometer.
By considering the solid angle subtended by the innermost ring and using
the small angle formula with equation (3), we arrive at the important relation
which leads to the more familiar form
This equation has been used to demonstrate that FabryPerot interferometers,
at a given spectroscopic resolution, have a much higher throughput than
more conventional techniques (Jacquinot 1954; 1960). But the Jacquinot
relation has another important consequence. The solid angle of a ring defined
by its FWHM intensity points can also be written
W = 2pqdq
= pl(\cal Nml)^{1} 

(6) 
For a fixed etalon spacing, the solid angle, and hence the spectroscopic
resolution, of all rings is a constant. This allows us to write down a
simple equation valid for all rings for the signaltonoise ratio in a
monochromatic unresolved line, viz.
SNR = s (eth)^{0.5} 
æ
ç
è 

dl^{¢
}dl 

ö
÷
ø 
0.5


æ
ç
è 

W
w 

ö
÷
ø 
0.5

(s+b+ f dl(dl^{¢}et)^{1}s^{2}_{R})^{0.5} 

(7) 
where s and b are the source and background flux (cts pix^{1}
s^{1}), e and t
are the efficiency and exposure times respectively, w
is the solid angle subtended by a pixeI. The quantity dl^{¢}
is the wavelength dispersion (Å pix^{1}) and f is the number
of CCD exposures combined to form the deep spectrum. The factor h
is discussed below. We normally choose to place the ring center at one
corner of the field for two reasons. First, it is always necessary
to tilt the etalon in order to throw ghost light out of the field (Section
8). Secondly, the factor (W/w)^{0.5}
in equation (2) now ensures that the spectroscopic sensitivity is constant
over most of the field. There will be an almost linear dropoff in sensitivity
at large offaxis angles (far corner from the optical axis) where the rings
become seriously incomplete.
An important characteristic of a spectrometer is its `spectral purity'
dl,
i.e. the smallest measurable wavelength difference at a given wavelength
(Thorne 1988). This is usually defined as the intensity FWHM of the instrumental
profile. When considering the amount of light transmitted by a spectrometer,
we need to consider the total area under the instrumental function. This
issue is rarely mentioned in the context of longslit spectrometers partly
because their response is highly Gaussian,^{1}
in which case the `effective photometric bandpass' (total area divided
by peak height) is very close to the FWHM of the instrumental profile (see
Table 1).
beam 



telescope diameter 
3910 mm 

central hole diameter 
1610 mm 

pupil size 
59.9 mm 

pupil stop diameter 
45.0 mm 

telescope area 
4.74×10^{6} mm^{2} 
etalon 



free spectral range 
57.0Å 

effective finesse 
50 

spectral purity 
1.15Å 

spectral sampling 
0.34Å 

effective bandpass 
1.80Å 
CCD 



gain 
2.7 e^{}/dn 

read noise 
2.3 e^{} 

pixel size 
24.0 mm 

pixel field 
0.594 
Table 2: TAURUS2
instrumental parameters for a recent observing run at the AAT 3.9m using
a 50mm diameter etalon. The effective telescope area is reduced by the
aperture stop.
At moderate finesse, the Airy function allows through 50% more light
than a Gaussian profile with equal spectroscopic resolution (Fig. 3). Thus,
the effective photometric bandpass dL is almost
60% larger than the bandpass defined by the profile FWHM dl.
The factor h in equation (7) corrects for a
calculation based on the FWHM of a ring and is defined as dL/dl.
Technically, dL should be adopted as the spectral
resolution of the Airy instrumental profile, otherwise we are forced to
a serious inconsistency when comparing FabryPerot spectrometers to other
devices (BlandHawthorn & Jones 1994).
4. Photometric calibration procedures
When we build up a data cube or take a series of observations, it is essential
to think of the scan variable as the etalon gap l rather than wavelength.
These
two variables should never be confused.^{2}
The wavelength range is moderated by the filter; the etalon gap is not.
The physical plate scanning range is l_{0} ±2Dl
= l_{0} ±2(l_{0}/2)
where l_{0} is the zeropoint gap and Dl
is the free spectral range in physical gap units. With this important distinction
in mind, for the flux in a standard star observation, we are able to write
S(l) = 
ó
õ 
F_{S}(l) \cal A(l,l)
dl 

(8) 
where F_{S}(l) is the product of the
stellar spectrum and the filter response. We can write down related expressions
for the calibrations. The limits of the integral in equation (9) are defined
by the bandpass of the entrance filter. The transform is some form of
a convolution equation in that \cal A(l,l) broadens
F_{S}(l) although, technically, the
term `convolution' should be reserved for integrals of the form
S(l) = 
ó
õ 
F_{S}(l) \cal A(ll)
dl 

(9) 
but note that this is a special case of equation (8). Suffice it to say,
a spectral line broadened by a spectrometer arises from a convolution and
not from a product^{3}.
4.1 Narrowband imaging
The Jacquinot spot is defined as the field about the optical axis within
which the peak wavelength variation with field angle does not exceed Ö2
of the etalon bandpass (Jacquinot 1954; Taylor & Atherton 1980). This
angular field can be used to perform close to monochromatic imaging. In
this section, we demonstrate how to convert the observed counts to true
flux units. We shall assume that the entrance filter selects a bandpass
which is a fraction of the free spectral range such that there is no pollution
from neighbouring orders.
We determine the TAURUS2 system efficiency by comparing the observed
counts for the flux standard h Hyades with the
expected counts (Hayes 1970) for which, at Ha,
we expect f_{l} = 2.65×10^{11}
erg cm^{2} s^{1} Å^{1} or equivalently
n_{l} = 8.69 phot cm^{2} s^{1}
Å^{1}. For the system parameters listed in Table 2, we observed
3.50×10^{4} counts for the flux standard in a one second
exposure. The efficiency calculation involves corrections for (A) effective
photometric bandpass (measured from the calibration data), (B) exposure
time, (C) CCD gain, (D) reduced telescope area, and (E) airmass. We now
quantify these stages:
A. 
3.50×10^{4} 
/ 
1.80 
= 
1.94×10^{4} 
cts Å^{1} 
B. 

/ 
1.00 
= 
1.94×10^{4} 
cts s^{1} Å^{1} 
C. 

* 
2.70 
= 
5.25×10^{4} 
elec s^{1} Å^{1} 
D. 

/ 
4.74×10^{4} 
= 
1.11 
elec cm^{2} s^{1} Å^{1} 
E. 

* 
1.03 
= 
1.14 
elec cm^{2} s^{1} Å^{1} 
It follows that the TAURUS2 instrumental efficiency is 13.1% or about
15.0±0.5% in the absence of the entrance
filter. Thus, we are now able to convert each recorded electron to exoatmospheric
flux units.
4.2 Spectral data cubes
The important point to realize here is that, with longslit spectrometers,
the instrumental profile is projected onto the detector. In contrast, with
scanning FabryPerot interferometers, we have the freedom to sample the
spectral line of interest however we wish in the scanning dimension. We
demonstrate the flux calibration of a spectral scan using the planetary
nebula flux standard IC 2165 (Lang 1980). With TAURUS2, a total of 1.50×10^{6}
counts was obtained from integrating over the Ha
line. The various stages are as before with three additional steps to correct
for (F) system efficiency, (G) sampling interval (the same filter was used
as before), (H) photon energy at Ha.
A. 
1.50×10^{6} 
/ 
1.80 
= 
8.33×10^{5} 
cts Å^{1} 
B. 

/ 
10.0 
= 
8.33×10^{4} 
cts s^{1} Å^{1} 
C. 

* 
2.70 
= 
2.25×10^{5} 
elec s^{1} Å^{1} 
D. 

/ 
4.74×10^{4} 
= 
4.75 
elec cm^{2} s^{1} Å^{1} 
E. 

* 
1.03 
= 
4.89 
elec cm^{2} s^{1} Å^{1} 
F. 

/ 
0.131 
= 
37.3 
phot cm^{2} s^{1} Å^{1} 
G. 

* 
0.34 
= 
12.7 
phot cm^{2} s^{1} 
H. 

* 
3.03×10^{12} 
= 
3.85×10^{11} 
erg cm^{2} s^{1} 
Our nebular flux 3.85±0.17 ×10^{11}
erg cm^{2} s^{1} compares remarkably well with Lang's
value of 3.86 ×10^{11} erg cm^{2} s^{1}
where we have assumed Case B recombination to convert from Hb
to Ha.
4.3 Diffuse source detection
A number of authors have exploited the Jacquinot advantage of the FabryPerot
to obtain extremely deep spectra of extended, diffuse objects. For a fixed
gap spacing, l µ
cosq, such that the spectrum in a narrow band
is dispersed radially from the optical axis across the field. When the
data are binned azimuthally about the optical axis, a single deep spectrum
is obtained. Like longslit spectrometers, the instrumental profile is
projected onto the detector but varies across the field according to dl
µ q^{1}.
At the AAT, we have already achieved Ha emission
measures of 0.2 cm^{6} pc (2×10^{19} erg cm^{2}
s^{1} arcsec^{2}) at the 3s
level in about 90 minutes. In principle, we are able to reach 0.02 cm^{6}
pc (3s) in about six hours using 3" optics.
The raw spectrum has quadratic sampling and needs to be resampled to a
linear axis where the original number of bins is preserved. As is evident
from equation (7), each pixel defines both a spectral interval (applied
in step G) and a projected solid angle. We include an additional step (I)
to correct for the latter, at which point the final spectrum has units
of erg cm^{2} s^{1} Å^{1} arcsec^{2}.
This procedure is to be discussed in more detail elsewhere.
Figure 4: Ghost
families arising from internal reflections within a FabryPerot spectrometer
(see text). (i) Diametric ghosts. Rays from the object O form an inverted
image I and an outoffocus image at R_{3}. The reflection at R_{1}
produces an outoffocus image at R_{2}. The images at R_{2}
and R_{3} appear as a ghost image G at the detector. (ii) Exponential
ghosts. The images at R_{2} and R_{4} appear as ghost images
G_{1} and G_{2} respectively.
5. Whitelight calibration
It is hard to overstate the importance of the whitelight cube. This is
obtained by observing a whitelight source over the same range of etalon
spacings used in the actual observations. The whitelight cube maps the
response of the filter as a function of position and etalon spacing. There
are three effects that we wish to divide out from the data. First, the
narrowband filter response, when convolved with the instrumental response,
leads to a modulation in the observed spectrum. Secondly, it is well known
that filters have variable responses in both collimated and converging
beams (Lissberger & Wilcock 1959). Thirdly, we seek to remove any inhomogeneities
in the filter structure as a function of position. Finally, when the whitelight
cube is compressed in the spectral dimension, it provides a very high signaltonoise
flatfield for removing pixeltopixel sensitivity variations.
Since it would be impractical to observe a flux standard at every pixel
position, we can only flux calibrate the spectral response at each point
in the field through the whitelight cube. Thus, we effectively calibrate
the whitelight response at the position of the flux standard and thereafter
the data cube. It is important to note that a twilight observation provides
a better model of uneven illumination across the field. Thus, we replace
the low frequency structure in each frame of the whitelight cube with a
single high signaltonoise twilight observation.
6. Lowlevel background effects
Besides the vignette problem, other important lowlevel effects are CCD
fringes and the `extraneous etalon' Airy pattern (see Section 8). The highfrequency
Airy pattern can have a similar amplitude to the CCD fringes. When using
narrowband entrance filters, we find that the fringe pattern for some chips
can show noticeable variations over a baseline as small as 10Å. This
may require coadding several whitelight cubes to divide out the fringe
pattern properly.
7. Atmospheric attenuation and seeing
An obvious limitation to good spectrophotometry is the degree of atmospheric
stability. The PEPSIOS system made use of a reference channel in order
to assess this (Hobbs 1969). With modern day imaging FabryPerot systems,
we use stars in the field. In good photometric conditions, we find that
the stellar intensities, and for that matter, the sky background, map the
filter structure rather well. Under these conditions, the CCD always outperforms
a photoncounting device (Bland & Tully 1989). In the presence of cirrus,
we often find that the sky continuum follows the filter structure while
the stellar response does not, and the stellar FWHM can be highly variable.
Such data are of limited use. In certain instances, it may be possible
to recover some measure of photometric integrity by spatially filtering
the data, particularly in conditions of variable seeing. In the early days,
TAURUS used the Image Proportional Photon Counting System to scan rapidly
and repeatedly to beat down these variations (Taylor & Atherton 1980).
To our knowledge, however, these data were never flux calibrated so it
is difficult to assess the possible gains.
Figure 5: Left. Two
ghost families are seen in this image of NGC 1068 taken with the Rutgers
FabryPerot on the CTIO 4.0m telescope. The optical axis is indicated by
the cross. `N' is placed slightly to the north of the Seyfert nucleus.
`DG1' is the diametric ghost of the active nucleus; `EG1' is an exponential
ghost of the Seyfert nucleus. `DG2' is the diametric ghost of `EG1' and
`EG2' is the exponential ghost of `EG1'.Right.The
`extraneous etalon' ghost pattern from the downstream etalon plate (lower
right quadrant) and from an air gap in the MOSFP camera (central). The
data were taken at the CFHT by illuminating the dome with an Ha
lamp. The fringe pattern of the Loral #3 CCD has a similar peak to trough
amplitude.
8. Ghost families
Even a minimal FabryPerot arrangement can have eight or more optically
flat surfaces. At some level, all of these surfaces interact separately
to generate spurious reflections. The periodic behavior of the etalon requires
that we use a narrowband filter somewhere in the optical path. Typically,
the narrowband filter is placed in the converging beam before the collimator
or after the camera lens, or in the collimated beam. The filter introduces
ghost reflections within the FabryPerot optics. The pattern of ghosts
imaged at the detector is different in both arrangements, as illustrated
in Fig. 4. Examples of these ghosts are shown in Fig. 5. A good way to
track these down is to place a regular grid of holes in focus at the focal
plane and illuminate the optical system with a whitelight source. We tilt
the etalon in such a way that the ghost images of the grid pattern avoid
the detector area.
Figure 6: The extraneous
etalon effect (see text).
A more difficult problem arises from the optical blanks which form the
basis of the etalon. These can act as internally reflecting cavities since,
from Section 2, if we let \cal R_{1} = 0.96 and \cal R_{2}
= \cal AR_{1} = 0.04 (airglass), we see that this generates a
ripple pattern with a finesse close to unity. The large optical gap of
the outer plates produces a highorder Airy pattern at the detector (Fig.
6). Traditionally, the outer surfaces have been wedgeshaped to deflect
this spurious signal out of the beam. Even curved lens surfaces occasionally
produce `halation' around point source images which may require experimenting
with both biconvex and planoconvex lenses when designing a focal reducer.
At the risk of belabouring the point, it is foolhardy to be using a FabryPerot
device unless the influence of scattered light is well understood.
9. Conclusions
In this review, we have outlined the main principles behind FabryPerot
spectrophotometry. We have also described issues which need to be addressed
if the photometric calibration is to be reliable. The imaging FabryPerot
interferometer has the capability to provide superior spectrophotometry
since slitaperture devices suffer seeing losses and narrowband filters
are tacitly assumed to have constant transmission properties as a function
of both position and wavelength. FabryPerot interferometers are still
not commonuser instruments at any observatory for a variety of reasons,
most notably because of the restricted wavelength coverage. But for studying
extended emission from a few bright lines, the capabilities of the FabryPerot
are unmatched by any other technique, with the exception of the imaging
Fourier Transform spectrometer (e.g., Maillard, this conference). Arguably,
the best results have been obtained when FabryPerot data are combined
with complementary data from narrowband and longslit devices (see the
contributions of Cecil, Hippelein, Pogge, Tully and Veilleux).
It is noteworthy that, in contrast to optical etalons, gapscanning
infrared etalons (e.g. Reay, this conference) have preceded the tremendous
ongoing advances in infrared detector technology. At optical wavelengths,
CCDs continue to improve their performance although there are outstanding
areas for future development (e.g., read noise, readout time). Thus, it
would seem that FabryPerot studies, particularly at infrared wavelengths,
hold great promise for the future. We only hope that future observing programmes
make full use of the spectrophotometric capabilities of the imaging FabryPerot
interferometer with proper attention to the issues discussed in this review.
I would like to thank the Marseilles Observatory for their excellent
hospitality. As always, I am indebted to my HIFI colleagues for their inspiration,
encouragement and friendship. Gerald Cecil assisted with Figure 4.
Atherton, P.D., Reay, N.K., Ring, J. & Hicks,
T.R. 1981, Opt. Eng., 20, 806 Bland, J. & Tully, R.B. 1989, AJ, 98,
723 BlandHawthorn, J. & Jones, A.W. 1994, preprint Hayes, D.S. 1970,
ApJ, 159, 165 Hobbs, L.M. 1969, ApJ, 157, 135 Jacquinot, P. 1954, J. Opt.
Soc. Amer., 44, 761 Jacquinot, P. 1960, Rep. Prog. Phys., 23, 267 Lang,
K.R. 1980, Astrophysical Formulae, SpringerVerlag, Berlin Lissberger,
P.H. & Wilcock, W.L. 1959, J. Opt. Soc. Amer., 49, 126 Roesler, F.L.
1974, Meth. Expt. Phys., 12A, chap. 12 Taylor, K. & Atherton, P.D.
1980, MNRAS, 191, 675 Thorne, A.P. 1988, Spectrophysics, Chapman &
Hall, London
I have always been of the view that wedging (e.g. Hernandez' text on
FabryPerot interferometers) is not so difficult to achieve since the outer
surfaces do not need to be optically flat. Making the plates optically
parallel doesn't help matters. I think the most practical solution is to
AR coat the outer surface of the downstream plate so as to beat down the
geometric mean of the reflectivities on both surfaces. An overall finesse
much less than unity suppresses the amplitude of the fringing. However,
the AR coatings with lowest reflectivity ($<$0.25
Footnotes:
^{1} The theoretical
sinc^{2} response of slitaperture devices is rarely achieved in
practice.
^{2} The
scan axis of a data cube becomes the l dimension
only very late in the reduction process.
^{3} This
is easy to verify analytically with two Gaussian functions.
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