Photometry and Polarimetry

1. Introduction

Most interpretation of the processes in an astronomical source rests on measuring its spectral energy

distribution (SED: generally defined as the distribution of output flux measured at low spectral

resolution over a broad range of the electromagnetic spectrum) or taking spectra (measuring the

distribution of output flux at high resolution). The spectral resolution is defined as R = where is

the wavelength of observation and is the range of wavelengths around that are combined into a

single flux measurement. Photometry refers to the procedures for measuring or comparing SEDs and is

typically obtained at R ~ 2 - 10.

In the optical and near infrared, nearly all the initial photometry was obtained on stars, whose SEDs are

a related family of modified black bodies with characteristics determined by a relatively small set of free

parameters (e.g., temperature, reddening, composition, surface gravity). Useful comparisons among

stars can be obtained relatively easily by defining a photometric system, a set of response bands for the

(telescope)-(instrument optics)-(defining optical filter)-(detector) system. Comparisons of

measurements of stars with such a system, commonly called colors, can reveal their relative

temperatures, reddening, and other parameters. Such comparisons are facilitated by defining a set of

reference stars whose colors have been determined accurately and which can be used as transfer

standards from one unknown star to another. This process is called classical stellar photometry. Its

validity depends on the stability of the photometric system and the accuracy to which it can be

reproduced by other astronomers carrying out comparable measurements.

Sources can also be monitored for variability using purely relative measurements without reference to

physical calibration. Such measurements are typically made in comparison to nearby sources thought

not to vary.

In other circumstances, it is necessary to convert measurements with a photometric system into

physical units. This situation applies in general for non-stellar targets and, since such targets are

dominant away from the optical and near-infrared, it is the rule for most wavelength regions. The initial

steps are similar to those for classical stellar photometry; a photometric system must be defined, a

network of standard sources is established, and measurements are made relative to these standard

sources. However, the outputs of the standards must be known in physical units, so the

intercomparisons can be expressed in such units. The photometric system must be based on spectral

bands covering a significant range of wavelength (to let signal through), but the specification of a SED in

physical units is in terms of a unit bandwidth (e.g., Hz or m). Corrections must be determined from the

nominal measured values to their equivalents at unit bandwidth.

Finally, although typical photometry refers to the total energy in a photon stream from a source, there

can be additional information in the polarization of this stream. It might seem that this information

could be extracted just by doing photometry through a polarizer, but the situation for accurate

measurements is a bit more complex, as we will discuss at the end of this chapter.

2. Getting the measurements

Historically, accurate photometry was conducted with single-detector instruments with properties optimized for this purpose. Accurate results depended critically on the quality of the observing conditions and the skill of the observer. Such instruments are still in use, but it is more likely that you will do photometry using an imaging detector. Given a well-reduced image, as described in the preceding chapter, photometric information can be extracted by routines available in a number of image analysis packages. They offer a number of approaches: Aperture photometry: a digital aperture is placed over the source image, the counts within the aperture summed, the aperture (or more often an annulus around the source position) is placed on nearby blank sky and the counts summed, and then the second sum is subtracted from the first (with appropriate weighting if the data are taken over different areas). This procedure is the same as is used with a single detector aperture photometer, but it is done in the computer after the data have been obtained.

� The aperture is placed on the basis of a centroiding routine, such as column and row sums or central moment, or by eye. Software such as DAOFIND can be used to locate all the sources in a field. � You must decide on the aperture diameter. It is not necessary to get nearly all the signal, so long as you are measuring point sources and do them all the same. In crowded fields you need to use small apertures. For faint sources, you will find there is a relatively small aperture size that optimizes the signal to noise even with no nearby sources. You should experiment with different aperture sizes to determine which is best for your application. In the end, you will correct for the signal lost from the aperture using measurements in multiple apertures on a bright, isolated source - this factor is the "aperture correction." � The "sky" will have deviations with a bias toward positive values (e.g., sources). You can use software to remove the obvious ones and replace them with mean surrounding values. In general, you may want to estimate sky as the mode, rather than median or mean, since the mode (most often occurring value) is most immune to bias.

Point spread function (PSF) fitting: The PSF is determined on a bright star and this profile is fitted to the images in the frame by least squares minimization. In crowded fields, the fitting can include many stars at once (DAOPHOT) to extract a best fit to the photometry of overlapping sources. Even for isolated sources, in principle PSF fitting should give the best signal to noise because it weights the pixel data more optimally than in aperture photometry.

� PSF fitting is more immune to anomalies than aperture photometry (e.g., a deviant pixel due to a poorly removed cosmic ray hit will just be counted at its measured value in aperture photometry, but will generally have a modest effect on the PSF fit to the image) � Calibration is preserved either with a PSF fit to a standard, or more commonly by comparing aperture photometry on an isolated bright source in your field with the PSF fit result. � Obviously, PSF fitting is not appropriate for extended sources, for which you need to use aperture photometry or an automatic program designed for extended sources such as Sextractor. NICMOS has provided a program very well suited for complex apertures on individual regions, IDP3.

Completing the job � Additional corrections may be needed at this stage: 1.) fixing cosNθ effects - you may need to have taken a "thousand points of light" measurement to get the correction (named in honor of President Bush I); and 2.) nonlinearity - your bright sources might be even a little brighter;

� In general, your reduction will be quoted in "instrumental magnitudes", and you will need to find what signals standards give in instrumental magnitudes to put the data on a standard scale. The necessary correction is sometimes called the zero point term.

You then need to carry out a series of standard procedures to correct for external (to the photometer) sources of error and to put your measurements on a standard basis, for example: 1.) air mass corrections; 2.) color corrections to put the data on a standard system. To carry out these last steps, it is required that you have measured standards or at least well behaved calibration stars over a large range of air mass. Color corrections may be available from prior measurements for your instrument, but it is desirable to measure standards over a range of colors to be sure they work well for your data. It can be challenging to reach accuracies better than the 1% level with imaging detectors used in standard ways. However, under special circumstances, imagers can do considerably better than they do for routine photometry - for example, where sources are always measured at the same positions and the images are well sampled (sometimes the images are put slightly out of focus to spread them over more pixels).

3. Stellar Photometry

For reasons that are arcane (to put a positive spin on them), astronomers conduct their

photometry in units of magnitudes, 2.5 times log base 10. To communicate with your

astronomical colleagues, you will need to do the same. Here is how to speak their language:

If your goal is to measure stellar colors in a relative sense, then you can proceed as follows:

Select a detector and a suite of filters Define some combination of reference stars that define “zero color” Measure a network of ‘standard stars’ of varying colors relative to these reference stars Measure stars relative to these standard stars

In general, m1 – m2 = -2.5 log (f1/f2), where m is the apparent magnitude Therefore, mB – mv = -2.5 log[fB/fb(0)]-{-2.5log[fV/fV(0)]} = -2.5 log[((fB/fV / (fB(0)/fV(0))], where fx(0) is the flux density through the x filter for a zero

color star (obviously, in the above fB(0) and fV(0) must be for the same calibrator star) Terminology

apparent magnitude, m = brightness of star in terms of m=-2.5 log (fstar/fstandard)+mstandard, normally at a specific wavelength or spectral band, e.g., mV for V magnitude

absolute magnitude, M = magnitude star would have at 10pc distance modulus, m-M = 5 log(dist in pc) – 5 = 5 log (d/10pc) bolometric magnitude, Mbol, is the absolute magnitude integrated over all wavelengths to give

luminosity bolometric correction, BC, is the quantity that must be added to the magnitude at some

wavelength to give mbol, the apparent bolometric magnitude color index, CI, is the difference between magnitudes in two different bands, e.g., CI=mB-mV,

often abbreviated as B-V color excess, E, is the difference between the observed CI and the standard CI for the stellar

type, e.g., EBV=CIstar-CIstellar type An example: “Arizona” UBVRIJHKLMNQ

Johnson UBVR system based on combinations of colored glass and the spectral responses of photomultipliers. I was an interference filter.

This system had the appearance of being cheaply and easily reproduced, but the detector-dependence resulted in a single “system-defining” photometer

The system was extended into the infrared with interference filters. The infrared bands are matched to the terrestrial atmospheric windows.

Figure 1. Johnson UBVRI passbands (normalized to unity).

H KJ

L MN

Q

Figure 2. Atmospheric transmission in the infrared under low water vapor (top curve) and

high water vapor (bottom curve). Where appropriate, the photometric bands are labeled.

Corrections for atmospheric absorption

“Air mass” refers to the amount of atmosphere along the line of sight, normalized to 1 when looking vertically

For a plane parallel atmosphere, air mass goes as

)1()sinsincoscos(cos)sec( 1 HAanglezenith

where is the latitude, HA is the hour angle at which the source is observed, and is its

declination.

This simple formula works remarkably well, with errors < 10% up to 4 airmasses and < 2% up to 2! A more accurate fit is

)2())1(sec0012.01(sec 2 zzairmass

where sec z can be obtained from equation (1).

Assuming the extinction is exponential,

)3(, I

ds

dI

then

)4().1(5.2)1()( amammamm

Establishing a system of standard stars With a photometer, a set of spectral bands, and methods to make accurate measurements (e.g.,

corrected for atmospheric extinction), we can compare stellar outputs accurately. We can not do nearly as well working with the straight signals from individual stars. Therefore, accurate stellar photometry requires a set of standard stars that we use as the foundation for comparative measurements.

Zero point - we have to pick some stellar reference that we are going to define to have "zero" color, and we will measure all other stars as blue or red relative to this reference. Johnson selected six A0 stars, and set Vega close to zero magnitude (for arcane reasons related to the five other stars, Vega was not set exactly to zero).

Extensive measurements were made in this system to establish the basic behavior of normal stars. This system is called the Arizona/Vega system of photometry. Harold Johnson made an enormous contribution to astronomy by developing a uniform and

accurate system of photometry (and instruments to implement it) from the optical through the

infrared.

Some difficulties in basic photometric systems We use the Arizona system just as an example. There are many others in the visible range, although in the infrared the locations of the atmospheric windows has impacted the ability of astronomers to invent totally new systems. Space observatories, with no constraints from the atmosphere, have introduced additional filter systems. We discuss below the steps required to place measurements made with one system onto a basis equivalent to measurements made with another system.

Reproducing the photometric system: The original Johnson UBVRI system depended for its

definition on a specific photometer - and when a detector failed, there was a crisis to find a replacement with the exact same spectral properties. By now, people have long since abandoned the idea of a defining photometer.

However, as a result each implementation even of the standard “Johnson” system may differ in subtle ways from others.

A further and larger difficulty arises in comparing radically different systems, for example the Sloan Digital Sky Survey u’, g’, r’, I’, and z’ with the Johnson bands

The figure below compares different versions of the Stromgren uvby system, compared with the

‘standard’ in the bottom panel

Figure 3. Realizations of the Stromgren “system.”

Serious trouble arises when transferring systems to CCDs, because of their intrinsically different spectral

response

When an exact equivalence cannot be found, one has to use a “color transformation”, for example:

v = V +const. + 0.1373 AM - 0.0160 (B-V)

to transform WYNN CCD v to Johnson V, where AM is the air mass at which the source is

observed.

Be aware that not all “equivalent” systems are! The filter set below was offered until recently by Barr Associates as "UBVRI." If you compare

with the Johnson system, you will see that R is far broader, I is moved to significantly shorter

wavelength, and there are smaller differences in other bands.

A few years ago Barr Associates and OCLI offered “astronomy” JHKLM sets, but they are no

longer on the market. You have always been on your own for Q, and N is somewhere in

between. There is also a popular “KS” filter that passes from 2.0 – 2.3m, based on the one used

in the 2MASS survey. Currently, obtaining filters often requires expensive custom coating runs,

Figure 4. Barr UBVRI filter set (no longer available).

or as in the very early days of infrared astronomy, searching the OCLI (now JDSU) stock filter

catalog. For this and other reasons, infrared filters with the same name may not have the same

bands!

"I" band is particularly problematic, with center wavelengths ranging by over 0.1m!

More Complications Atmospheric transmission in the infrared; the photometric system is matched to the

transmission windows, but the spectral bands still depend on atmospheric absorption as well as on the filter and detector characteristics: for examples, look at the ‘M’ and ‘Q’ bands for which the figure above gives transmissions for differing amounts of water vapor. Similarly, the U band shape is subject to atmospheric effects.

Systems may have filter leaks

System properties may be temperature dependent The figure below shows the temperature shifts in a detector spectral response.

Figure 5. Red leaks in the Johnson U and B bands.

Figure 6. Detector spectral response vs. temperature

Figure 7 shows temperature effects on interference filter passbands. Thus, two systems using

identical detectors and filters but at different temperatures will have different spectral properties.

What's a gal or guy to do? Transformations

Photometric systems that are stable and have been used extensively can be compared against

measurements of stars in other systems. These comparisons need to extend over the full range

of stellar colors to be generally useful.

Below is an example (from Carpenter 2001, AJ, 121, 2851) comparing 2MASS with an earlier

photometric system introduced by the Caltech IR group (CIT). The fitted transformations are:

)5()005.0028.0())(020.0026.1()(

)005.0013.0())(006.0056.1()(

)006.0043.0())(010.0076.1()(

)003.0024.0())(005.0000.0()(

2

2

2

2

CITMASSS

CITMASSS

CITMASS

CITCITMASSS

KHKH

KJKJ

HJHJ

KJKK

Figure 7. Interference filter wavelength shift vs. temperature.

The corrections are fairly large. The lower panels in the figures show the residuals, which are

small (but based on a worrisomely small sample of stars). Therefore, application of the

transformations can homogenize the photometric systems reasonably well so long as they are

being applied to normal stars, since the transformations refer to such stars.

However, if you are doing very demanding work, be suspicious. For example, there are inconsistencies within 2MASS between bright and faint sources at the 2% level.

New homogeneous all sky databases (2MASS for NIR, Hipparcos for BV) make the situation much better than it was in the recent past.

3. Physical Photometry

Figure 8. Photometric transformations, 2MASS to CIT.

Classical stellar photometry is of little use when studying something other than a relatively normal star,

since it assumes a related family of pseudo-blackbody spectra. In addition, the underlying precepts of

classical photometry break down at wavelengths where stars are not readily measured. Thus, physical

photometry – direct conversion to real physical units – is the assumed approach in the radio, far

infrared, and X-ray. In the optical and infrared, however, we have to define the approaches to be used in

distinction to those for classical stellar photometry.

A physical system converts the measurements into flux densities at specific wavelengths

To each filter, we assign an effective wavelength, 0. There are a number of possible definitions, but for now we will use the mean wavelength:

)6(.)(

)(

0

dT

dT

T() is the wavelength dependent system transmittance (including filter, detector, telescope,

and atmosphere). See Figure 9.

We will use our measurements to determine the flux density of sources at this effective wavelength. To do so, we need an "absolute calibration" - that is, we need to know accurately the flux densities from our standard stars in correct physical units.

Figure 9. Photometric passband and effective wavelength.

Absolute Calibration Physical photometry is carried out relative to standard stars or other sources, just as in classical

photometry. The difficulty is to know their correct flux densities. There are a number of possibilities

to get this information.

Direct calibrations One transfers a calibrated blackbody reference source to one or more members of the standard star

network. Ideally, one would use the same telescope and detector system to view both, but often the

required dynamic range is too large and it is necessary to make an intermediate transfer.

Indirect calibrations One can use physical arguments to estimate the calibration, such as the diameter and temperature

of a source. A more sophisticated approach is to use atmospheric models for calibration stars to

interpolate and extrapolate from accurate direct calibrations to other wavelengths.

Hybrids The solar analog method uses absolute measurements of the sun, assumes other G2V stars have

identical spectral energy distributions, and normalizes the solar measurements to other G2V stars at

some wavelength where both have been measured, such as mV

Current "best" methods The calibrations in the visible are largely based upon comparisons of a standard source (carefully

controlled temperature and emissivity) with a bright star, often Vega itself. Painstaking work is

needed to be sure that the very different paths through the atmosphere are correctly compensated.

In the infrared, there are three current approaches that yield high accuracy:

Measurement of calibration spheres by the MSX satellite mission and comparing the signals with standard stars. This experiment has provided the most accurate values.

Measurement of Mars relative to standard stars while a spacecraft orbiting Mars was making measurements in a similar pass band and in a geometry that allowed reconstructing the whole-disk flux from the planet.

Solar analog method, comparing new very accurate space-borne measurement of the solar output with photometry of solar-type stars.

Overall status Accurate (better than 2% estimated uncertainty) calibrations are now available for the range 0.3

through 25m. They are usually expressed as the flux density corresponding to zero magnitude at

each effective wavelength, sometimes called the ‘zero point’. As a result of Johnson's work, the

zero point is defined in terms of Vega.

Thus, accurate physical measurements can be deduced over this range by measuring signals relative

to standard stars, correcting for atmospheric extinction, and then applying the calibration directly to

the measured magnitudes.

Some mischief Following astronomical tradition, Vega was a very bad choice for a star to define photometric

systems

It has a debris disk that contributes a strong infrared excess above the photosphere, already

detected with MSX at the ~ 3% level at 10m and rising to an order of magnitude in the far infrared

Interferometric measurements at 2m show a small, compact disk that contributes ~ 1% to the total flux

Vega is a pole-on rapid rotating star with a 2000K temperature differential from pole to equator.

This joke nature has played on Harold and the rest of us accounts for some of the remaining

discrepancies in absolute calibration.

Bandpass Corrections The signal is proportional to the convolution of the source spectrum with the instrument

spectral response function (see Figure 10 for an example)

Figure 10. Comparison of measurements of a hot star and a cool star in the B filter

Sources with the same flux densities at 0 but different spectral shapes may give different signals –

and conversely, sources giving the same signals may not have the same flux densities at 0. With

knowledge of the source spectrum, one can make corrections by integrating it and the standard star

spectra over the passband and comparing the results. Such a calculation is illustrated for the

HKLMNQ bands below. For clarity, each band is offset upward by 0.05 magnitudes, starting with N.

The correction is set to zero for stars, under the assumption that they are used as the standard

sources. If one has deduced a flux density at 0 in the conventional way by measuring a source of

temperature T relative to these standard stars, the best estimate of the true flux density at 0 is

obtained by subtracting the quantity graphed in the figure.

For example, any source so cold that the peak of its blackbody falls at longer wavelengths than the

system passband has a spectrum rising very steeply (Wien side of blackbody) across the passband.

Most of the signal will be derived from wavelengths longer than 0, and the flux density at the

center of the passband can be small even with a significant net signal. A correction in the opposite

direction is required when the blackbody peaks near 0.

Figure 11. Correction to equivalent monochromatic magnitude as a function of blackbody

temperature for the infrared bands H, K, L, M, N, and Q. The curves are offset by 0.05 successively

for clarity (Q to H).

The size of the corrections scales as (/0)2. The correction is moderate in size for /0 < 0.25, but

for broader filters the uncertainties in the correction can dominate other uncertainties in the

photometry.

An example of how to do all this is attached as an appendix.

(Misguided) attempts to minimize bandpass corrections

Some use alternates of 0: for example, IRAS defines

)7(.)(

)(2

dT

dT

eff

This definition reduces the corrections for warm and hot objects and increases them for cold

ones. It is harmless except for causing some confusion

A much worse approach is embodied in the isophotal wavelength. The idea is not to adjust the measured flux density, but to adjust the wavelength of measurement for every source so the measured flux density applies at that wavelength. This process is mathematically equivalent to adjusting the flux density, but has the unfortunate result that sources measured with the same

photometric system all have different wavelengths assigned to the results. Since 0 is one of the succinct ways to characterize the passband, the result borders on chaos (think of how to put the data into a sensible table!). Furthermore, real stars have absorption features, and so the definition of isophotal wavelength has to include interpolating over them to get an equivalent continuum. If the interpolation is done in different ways, one can get different isophotal wavelengths for the same measurement on the same star!

While we are discussing peculiar thought patterns, we have to mention “AB magnitudes”. These

take the zero magnitude flux density at V and compute magnitudes at all bands relative to that flux density. Thus, they are a form of logarithmic flux density scale, with a weird scaling factor of –2.5 and a weird zero point of ~ 3630 Jy. (This type of foolishness has led to mistakes causing waste of many orbits of HST time -- due to confusion between Johnson and AB magnitudes: mK(AB)-mK(Johnson)~2, for example.)

In case you need to use mAB to communicate with other astronomers, it is

)8(085.56))(log(5.2 12 HzmWfmAB

4. Other Types of Photometry

4.1 Differential Photometry ‘Spatial differential’: variability can be measured by monitoring nearby stars simultaneously with the

measurement target. This gain comes ‘for free’ with imagers, but can be extended to aperture photometers using more than one detector, for example. Atmospheric effects, filter leaks, and time variability of the system can all be made to cancel to first order.

‘Spectral differential’: narrow spectral features can be measured by comparing signals from a narrow and broad filter, both centered on the feature (there are other strategies too).

High speed photometry

With CCDs, the readout time is fixed by the speed of the transfer of signals (and the resulting charge transfer efficiency), plus the speed of the output amplifier and the receiving electronics; infrared arrays have similar limits if the entire array is read out

However, with nearly all arrays, one can just get signals from a subsection, which can be read out at the optimal speed per pixel. If only a small fraction of the pixels in the array are being read out, then this subsection can be read much faster than the entire array while still maintaining the good performance (for the subsection only, of course)

For example, a CCD can have the clocks run fast to shift the charge out over the "unwanted" area, dumping it by resetting. When you get to the subarray, the clocking is slowed down and the pixels are read out at the standard rate. When you get past the subarray, it is usually desirable to clock through the entire array so charge does not build up and bleed into regions where you do not want it.

In theory infrared arrays allow full random access - you could address just the subarray. However, since usually shift registers are built in to shift the entire array out serially, often one has to use a fast-advance strategy similar to that with the CCD to get to a subarray, and a similar readout strategy also.

A few arrays have extra logic to allow direct addressing of a subarray (e.g., the Teledyne arrays for NIRCam).

Equivalent width, W, vs ratio of signals

Figure 11. Photometric passbands for measuring Hline strength (absorption of emission) to

left, and to the right a comparison between spectroscopically measured equivalent widths of

the line and the ratio of signals in the two photometric bands.

One can also use a custom-manufactured small array if one wants fast readout without the advancing over pixels.

High speed photometry with detectors with gain

Another way to get high speed is to use gain so the effective read noise is kept small

CCDs are now available where the voltage on the output register is increased so there is mild avalanching, so as the charge is passed from one gate to another the amount of charge grows. Since the fast transfers in a CCD are all in the output register, this strategy lets one move the charge quickly and get enough signal that the residual extra noise from the degradation in CTE is not a problem. It also allows faster operation of the output amplifier.

Avalanche gain can also be used in silicon PIN diodes that are appropriately designed. These devices are used in a photon-counting mode, and for low enough input photon rates can have noises of just a few electrons. With higher input rates, the avalanche gain adds considerably to the noise, however.

In the infrared, the solid state photomultiplier (essentially an impurity band conduction (IBC) detector) can be modified in construction to have a gain region. With appropriate adjustment of both the bias voltage and the temperature of the detector, this device gives an avalanche-assisted large pulse out for each photon detected, analogous to the PIN avalanche pulse counting photodiode.

High accuracy (e.g., planetary transits)

Array detector virtually required because an aperture would modulate the signal with tiny telescope motions

Can mitigate pixel non-uniformity by putting many pixels across the point spread function.

Accurate placement of the star on the array and holding it at the same position also improves the relative photometry, since it prevents many of the noise sources due to intrapixel sensitivity variations and similar causes.

Analysis of the images to minimize systematic effects is critical. One effective technique is to convolve a template image of the field with the seeing at the time of observation and then scaling the template to match the image and subtracting the two. The photometry can then be performed on the residual image (Alard 2000). Other methods have also been used – simple aperture photometry works reasonably well -- and further progress is to be expected given that the success of transit photometry is dictated by the quality of the reductions.

Transit systems can be based on small telescopes (e.g., telephoto lenses for cameras) to study bright

stars.

Space telescopes are also used for transits very successfully. Although the many-pixel rule cannot be imposed, high quality data are being obtained by careful modeling of the results, taking advantage of the steady pointing and lack of effects from the atmosphere.

5. Polarization

5.1 Background Electromagnetic radiation consists of transverse vibrations of the electromagnetic field. If they are incoherent, such vibrations can separately propagate through space without change and without interfering with each other. If the phase and amplitude of these separately propagating vibrations have no fixed relationship to one another, it is said that the radiation is unpolarized. If there are lasting relationships in phase and amplitude, then the light is at least partially polarized.

Figure 12. Planetary transit observation.

Processes that lead to significant polarization include:

Reflection from solid surfaces, e.g., moon, terrestrial planets, asteroids

Scattering of light by small dust grains, e.g., interstellar polarization

Scattering by molecules, e.g., in the atmospheres of the planets

Scattering by free electrons, e.g., envelopes of early-type stars

Zeeman effect, e.g., in radio-frequency HI and molecular emission lines

Strongly magnetized plasma, e.g., white dwarfs

Synchrotron emission, e.g., supernova remnants, AGN The electric vector of a totally polarized beam of light can be described as

)9()cos()cos( 00

jtEitEjEiEE yyxxyx

The tip of the electric field vector traces an ellipse on a plane that is perpendicular to the wave propagation direction (see Figure 13). The four Stokes parameters are used to describe this behavior. They are total intensity I, linear polarization given by U and Q, and circular polarization V. In general, polarization should be viewed as the combination of all four, in which case it is elliptical, with an intensity of

)10(222 VUQIPE

where PE is the degree of polarization (< 1). The Stokes parameters are defined in terms of the electric vector from equation (9) as:

)11(2sinsin2

2sin2sin2coscos2

2cos2cos2cos22

22

VEyx

Eyx

Eyx

yx

IPIPEEV

IPIPEEU

IPIPEEQ

EEI

where

)12(2cos EPP

is the degree of linear polarization and

)13(2sin EV PP

Figure 13. Electric vector ellipse

is the degree of circular polarization, positive for right-handed and negative for left-handed; that is, β

defines the relative amounts of linear and circular polarization. In addition, = x - y is the phase difference between the x and y vibrations.

As applied in astronomy, the angles and intensity determine a polarization vector in spherical coordinates. θ is the angle between the long axis of the polarization ellipse and the direction toward the north celestial pole (NCP), measured counter-clockwise from the direction toward the NCP. θ is called the position angle of the polarization vector. The factor of two before θ arises because a polarization ellipse is degenerate for rotations of 180o, while the two before β reflects a similar degeneracy for a 90o rotation of an ellipse accompanied by swapping its major and minor axes.

Conversely, if we can measure the Stokes parameters, we can solve for the polarization vector as

)14(arctan2

22

Q

U

I

UQP

I

The Stokes parameters are a convenient way to describe polarization because, for incoherent light, the Stokes parameters of a combination of several beams of light are the sums of the respective Stokes parameters for each beam. Now imagine that we have a polarization analyzer, a device that subdivides the incident light in half, with one beam linearly polarized in a direction we define as the principal plane of the analyzer and the other beam polarized in the orthogonal plane. We designate the angle between the north celestial pole and the principal plane as φ. When light characterized by the Stokes parameters I, Q, U, and V falls on the analyzer, the intensities of the two beams emerging from it are (Serkowski 1962)

)15()2sin2cos(22

)2sin2cos(22

2/12/1

2/12/1

UQTT

ITT

I

UQTT

ITT

I

rlrlOP

rlrlPP

where Tl is the transmittance of unpolarized light (e.g., measured with two identical analyzers oriented

with their principal planes parallel) and Tr is the transmittance of two analyzers with their principal planes perpendicular. For simplicity, assume a perfect analyzer, Tl = 0.5 and Tr = 0. Then

)16()2sin2cos(2

1

)2sin2cos(2

1

UQII

UQII

OP

PP

Thus, measuring the two beams emergent from the analyzer at a number of angles relative to

direction to the north celestial pole, φ, gives a set of values of IPP and IOP that can be solved for

Q and U. I can be measured as just the total flux from the source. Circular polarization is conventionally measured by inserting into the beam ahead of the analyzer an optical component that converts circular to linear polarization and then measuring the strength of this linear polarization.

5.2 Optical elements 5.2.1 Analyzers

Conceptually, perhaps the simplest form of analyzer is a fine grid of parallel conductors, placed perpendicular to the beam of light. If the spacing between wires is five times their diameters, then the grid acts as an efficient analyzer for wavelengths longer than about five times the grid spacing. Waves with electric fields perpendicular to the grid lines cannot transfer much energy to the grid because the grid diameter is so small compared with the wavelength, so they are able to travel through the grid. However, if the electric fields are parallel to the grid, the photons are reflected (or absorbed) efficiently. Given current capabilities to produce fine grids of conductors by photo-lithography, such devices work well throughout the infrared and can be made to work reasonably well in the visible. A similar concept explains the operation of sheet polarizing material, used in the visible. Sheet analyzers can be fabricated in polyvinyl alcohol plastic doped with iodine. The sheet is stretched during its manufacturing so the molecular chains are aligned, and these chains are rendered conductive by electrons freed from the iodine dopant. Another class of analyzer is based on the birefringence of certain crystals – that is, in certain orientations these crystals have different indices of refraction for light polarized in different directions. Examples include magnesium fluoride, calcite, sapphire, and lithium niobate. A simple analyzer can be made with a plane-parallel calcite plate in front of the telescope focal plane, producing images in orthogonal polarization directions separated by 0.109

times the thickness of the plate (at 0.55m). However, because of the differing indices of refraction, the two images come to foci at different distances behind the plate. This issue can be avoided by using two (or more) plates of material. A well-known example is the Wollaston prism (Figure 14). It consists of two triangular right prisms cemented (or contacted) along their bases and with their polarization axes oriented orthogonally. When it encounters the interface between the prism, a beam of light finds that one polarization direction is transitioning from a medium with a relatively high refractive index to one with a relatively low one, and the other direction is transitioning from low to high. The result is that the two polarizations are deflected in opposite directions. Wollaston prisms can divide the two polarizations by 15 to about 45o in this fashion. There are a large number of alternative arrangements that perform functions similar to that of the Wollaston prism, e.g., the Glan-Thompson prism transmits one polarization undeviated but absorbs the other. 5.2.2 Retarders/wave plates

Figure 14. Wollaston prism (from Wikipedia)

If one direction of polarization is retarded in phase by 180o relative to the other, the direction of the polarization is rotated, as shown in Figure 15 (see also equations 11 and 14). A simple retarder can be made of a plane-parallel plate of birefringent material, suitably cut and oriented. The specific case of a 180o phase shift is called a half-wave-plate. In general more complex realizations of half-wave-plates use multiple materials to compensate for color differences and give a relatively achromatic phase shift.

By symmetry, a half-wave-plate with its optical axis oriented in the direction of polarization does not rotate the plane of polarization of the wave. If the optical axis is 45o from the plane of polarization, the half-wave-plate rotates the polarization 90o (see Figure 15), if the optical axis is 90o off the plane the rotation is by 180o (i.e., by symmetry, we are back to the beginning – no effective rotation).

Another special case applies when there is a 90o phase shift – then a perfectly plane polarized wave becomes perfectly circularly polarized (see Figures 15 and 16, and equation 11). Thus, a quarter-wave-plate converts linear polarized light to circular polarization, or circular to linear. There are other types of retarder besides passive plates of birefringent materials. One example is the Pockels cell. This device is based on materials in which the birefringence can be altered by an electric field, such as lithium niobate or ammonium dihydrate phosphate (AD*P). Birefringence is also induced in normal optical materials by mechanical strain. This effect is used in photoelastic retarders, which vibrate a piece of material at its natural resonant frequency to provide rapidly periodically varying retardation (e.g., at 20 kHz).

Figure 15. Action of a half-wave retardation to rotate the plane of polarization. (from Wikipedia)

5.3 Polarimeters A very simple polarimeter could be based on placing an analyzer in the beam and making measurements at different rotation angles of the analyzer. However, such an arrangement would have limited accuracy. A fundamental problem is that the measured effect appears synchronously with the rotation of the analyzer – any flaw in the optical transmission of the optical train from analyzer to detector will appear as a false polarization signal. In addition to wandering of the beam due to optical imperfections in the analyzer, the optical train behind it may transmit different polarizations with different efficiencies – for example, reflections off tilted mirrors induce polarization in the beam. Although it is possible to calibrate such false signals and remove them to some degree from the measured values, it is much better to use a polarimeter design that is less susceptible to them. One such design is to rotate the entire instrument, rather than just the analyzer. In that case, the path through the instrument remains identical for the different angles and the effects of polarization-sensitive optics should not matter. However, the instrument can not generally be rotated very fast, so any changes in conditions (e.g., seeing or atmospheric transmission) can have a large influence on the measurements made either at both different polarization angles or at significantly different times. Still, rotating the instrument can be a useful technique to probe possible instrumental errors. More adaptable designs are based on placing a waveplate or other retarder in the beam before it encounters any tilted mirrors. For a normal Cassegrain or Gregorian telescope, the two first reflections (primary and secondary) are nearly normal to the mirror surfaces and thus, by symmetry, induce very little polarization – the retarder is placed in the beam formed by these two mirrors. An analyzer is placed behind the retarder, and can in fact be behind any number of tilted mirrors or other polarizing optical elements without significantly compromising the performance. In fact, conceptually we can consider such elements just to be part of the action of the analyzer. Suppose that the polarization measurement is obtained by measuring the signals at different rotation angles of a half-wave-plate. The rotation of the angle of polarization will be 2θ for a rotation of the wave plate by θ, that is, changes in

Figure 16. (a) two electric waves π/2 out of phase and

(b) their superposition produces a circularly polarized

wave, from

http://webpages.ursinus.edu/lriley/courses/p212/lect

ures/node25.html

signal due to optical imperfections in the wave plate will have a different rotational dependence from changes due to a polarized input beam. These two improvements: 1.) immunity to polarization effects and other flaws in the instrument optics; and 2.) separating the signal from one-to-one correspondence with the motion of the analyzer, together yield a big gain in the accuracy with which polarization can be measured. A further gain arises because the wave plate can be moved to a new angle quickly, so different measurements can be obtained over time intervals short compared with changes in atmospheric transmission, seeing, and other variable effects. In addition, if the retarder is a Pockels cell or photoelastic device, there is no bulk mechanical motion associated with the rotation of the plane of polarization and in theory the polarization artifacts associated with the rotation of a conventional wave plate are avoided. High quality instruments have been built using any of these three approaches. In fact, placing the retarder ahead of the rest of the instrument and directly in the beam from the telescope allows high quality polarization measurements to be made with a variety of instruments, such as single beam photometers, imagers, and spectrometers. {footnote: Linear algebra is often used to analyze the behavior of a series of polarimetrically-active optical elements, each of which is described by a “Mueller

Figure 17. A spectropolarimeter (SPOL)

matrix.” The basics of polarimetry can be explained without this formalism, but anyone planning to get more familiar with the field should expect to encounter it.} 5.4 Measurement interpretation and error analysis

From equation 16, if we make a measurement of the polarization in the principal plane and in the orthogonal plane of our analyzer when it is at angle φ relative to the NCP, then

)17(2sin2cos

I

UQ

II

IIR

OPPP

OPPP

Although this expression assumes a perfect analyzer, it can be generalized to a more realistic case in a straightforward way. For φ=0, we get R0 = Q/I = q, while for φ=45o, we get R45 = U/I = u. We can adapt equations (14) for the amount and angle of polarization.

)18(arctan2

1

22

q

u

uqP

It is convenient to use a diagram of q vs. u, with angles in 2θ, to represent polarization measurements (Figure 18). For example, different measurements can be combined vectorially on this diagram. To see how this works, suppose we made two measurements of a source, the first giving 2% polarization at an angle of 20o and the second 2% polarization at an angle of 110o. These measurements are consistent with the source being unpolarized. If we put them on the q – u diagram and combine them vectorially, we indeed get a result of zero. Error analysis for polarimetry is generally straightforward, except when it comes to the position angle for measurements at low signal to noise. Assume that the standard deviations of q, u, and P are all about the same. Then the uncertainty in the polarization angle is

)19()(

65.28)(P

P

Thus, nominally a measurement at only one standard deviation level of significance (that is, a non-detection) achieves a polarization measurement within 28.65o. This high accuracy is non-physical – the probability distribution for θ at low signal to noise does not have the Gaussian distribution assumed in most error analyses (e.g., Wardle and Kronberg 1974). Similarly, P is always positive and hence does not have the Gaussian distribution around zero assumed in normal error analysis.

References

Figure 18. The q – u diagram.

H. L. Johnson 1966, “Astronomical Measurements in the Infrared,” in Ann Rev. Ast. & Astrophysics, 4,

193: classic discussion of photometry of normal stars in UBVRIJKLMN system

A. G. Davis Phillip (ed.) 1979, “Problems of Calibration of Multicolor Photometric Systems,” (Dudley

Obs., Schenectady, N. Y.): All you wanted to know (and more) about various photoelectric

photometric systems.

M. Golay 1974, “Introduction to Astronomical Photometry,” (Reidel: Dordrecht): Excellent, thorough

text on photoelectric photometry, although now a bit dated.

A. T. Young 1974, “Photomultipliers, Their Cause and Cure," “Other Components in Photometric

Systems,” and “Observational Technique and Data Reduction,” in Methods of Experimental Physics,

Vol. 12, Part A, Astrophysics, ed. N. P. Carleton: very thorough, advanced discussion of photometry; if

beginners start with these references, they’ll take up spectroscopy instead.

F. J. Low and G. H. Rieke 1974, “The Instrumentation and Techniques of Infrared Photometry,” in

Methods of Experimental Physics, Vol. 12, Part A, Astrophysics, ed. N. P. Carleton: Detector advances

make some of this material very dated, but it also gives the original discussion of IR optimization of

telescopes, and the basics of the IR photometric system.

C. Buil 1991, “CCD Astronomy: Construction and Use of an Astronomical CCD Camera,” (Richmond

VA: Willman-Bell): somewhat amateur-oriented description of use of CCD detectors, from basics of

operation through hardware and software and image reduction and analysis.

G. H. Jacoby 1990 (ed.), “CCDs in Astonomy,” Ast. Soc. of Pacific Conf. Series, 8 (San Francisco: ASP):

conference proceedings that give overview of current research in CCDs and the applications of these

devices.

Chr. Sterken and J. Manfroid 1992, “Astronomical Photometry, A Guide,” (Dordrecht: Kluwer):

probably best modern text on the subject.

Budding and Demiscan, Introduction to Astronomical Photometry 2nd edition, 2007 (Cambridge,

Cambridge Press): Another modern overview.

Howell, S. B. 1992, “Astronomical CCD Observing and Reduction Techniques,” (San Francisco: Ast.

Soc. of the Pacific): modern conference on usage of CCDs in imaging and photometry.

Tinbergen, Jaap 1996, “Astronomical Polarimetry,” New York, Cambridge: Cambridge University

Press

Serkowski, K. 1962, “Polarization of Starlight,” Adv. Astr. Astrophys., 1, 289

Wardle and Kronberg 1974, ApJ, 194, 249

Example of Photometric Reductions:

Band

Vega 1 Volt = 1000 ADU 0.5V = 500 ADU

Star X 0.015V = 15 ADU 1V = 1000 ADU

sky 0.010V = 10 ADU 0.025V = 25 ADU

Assume Vega has mJ = mK = 0.02

Compute mJ, mK, mJ – mK for star X

Compute the flux densities corresponding to the mJ’s and mK’s.

Note: from Campins et al. 1985:

mJ = 0 at 1603 Jy = 1.603 x 10-23 W m-2 Hz-1

mK = 0 at 667 Jy = 6.67 x 10-24 W m-2 Hz-1

1. We compute the sky-corrected signals:

Band

Vega 1000 – 10 ADU = 990 ADU 475 ADU

Star X 5 ADU 975 ADU

2. Assume the output ADUs are linearly related to the detected flux density:

ADU1 = C F1

ADU2 = C F2

Then the magnitude difference between star1 and star2 is

m12 = -2.5 log (F1/F2) = -2.5 log (ADU1/ADU2)

or, in our case,

mJ(X) – mJ(Vega) = -2.5 log (5/990) = 5.74

Since mJ(Vega) = 0.02, mJ(X) = 5.76

Similarly, mK(X) = -0.76

For star X, mJ – mK = 6.52.

3. To compute flux densities, start with mJ. By definition, the J flux density is fainter than that of a 0

magnitude star by

Thus, its flux density is

Similarly, star X has an indicated flux density of 1343 Jy at K.

4. Suppose we fit a power law to the spectrum of star X:

Then we can determine the index, , as

1.2015.2/76.510 E

2.12261098.798.7201

1603EHzWmJy

Jy

. 3F C E

and we find C = 1.018 X 10131.

Thus, the spectrum of star X looks roughly like

while that of the calibration star, Vega, is roughly Rayleigh Jeans,

5. Are E5 and E6 correct? We should worry about the possible influence of the strongly different spectra

on the measurement process. What we really measure is the total energy passed through the

photometric band-defining filter. To illustrate, approximate the K band as a square transmission

function, with cuton at 1.92m and cutoff at 2.52m. Then we can integrate across this function to get

the power through the filter for star X:

where 1=c/2.52x10-6m and 2=c/1.92x10-6m.

For Vega, we get by a similar integration,

Therefore, we predict a ratio of signals, starX/Vega, of 2.47, whereas we got 975/475=2.05.

Note that we have not been rigorous in terms of the overall system response, etc. – the example is only

illustrative.

4.047.9246.0

226.2

)2

/1

log(

)1

/2

log(E

FF

5.05.913110018.1 EF

6.226105485.3 EF

7,2101022.62

1

05.9 EWmdCP

8.21010516.2 EWmP

We conclude that the differing shapes of the spectra in the filter bandpass have biased our assignment

of the flux density of star X relative to that of Vega at the effective wavelength of the filter. To put them

on the same basis, we should multiply the derived flux density of star X by 2.05/2.47 = 0.83, while

retaining the same effective wavelength. That way, the integrals will correspond to the observed ratio.

Strictly speaking, we should then recompute the spectral shape and check the signal ratios again. Also, a

bandpass correction may be required at J.