Astronomical polarimetry is currently drawingmuch attention, mostly due to the anticipated high-sensitivity searches for the so-called B modes of thecosmic microwave background (CMB) polarization.These signatures of gravitational waves producedduring the inflationary epoch are expected to pro-vide a direct measurement of the energy scale ofinflation. The amplitude of the B modes is theorizedto be 10�7 to 10�9 of the power of the CMB, and so itsmeasurement will require a good modulation strat-egy and control over systematic artifacts.
The emission from magnetically aligned dust in ourgalaxy provides a contaminant that will have to beunderstood in order to correctly extract the B modesfrom the total signal. On the other hand, this polar-ized emission provides a tool for analyzing the role ofmagnetic fields in star formation, and with the ad-vent of multiwavelength submillimeter and far infra-red photometers such as the submillimeter common-use bolometer array (SCUBA2)1 and the highangular-resolution widefield camera/StratosphericObservatory for Infrared Astronomy (HAWC�SO-
FIA),2 there is an opportunity to expand this field ofstudy. To take advantage of the new detector tech-nology that will be coming online in the next fewyears, it is necessary to develop the polarization mod-ulation technology that will enable the conversion ofthese photometers into polarimeters.
Fundamentally, partial polarization arises as a re-sult of statistical correlations between the electricfield components in the plane perpendicular to thepropagation direction. These correlations are repre-sented by complex quantities, but in the measure-ment of polarized light, it is often convenient to usereal linear combinations of these correlations,namely, the Stokes parameters, I, Q, U, and V.
It is possible to trace the polarization state of ra-diation through an optical system by determining thetransformations that describe the mapping of the in-put to output polarization states. We are specificallyconcerned with the class of optical elements for whichStokes I is decoupled from the other Stokes parame-ters. For this class of elements, the polarization,
P2 � Q2 � U2 � V2, (1)
is constant. This equation can be interpreted to de-scribe the points on the surface of a sphere in a 3Dspace having Q, U, and V as coordinate axes. Thissphere is known as the Poincaré sphere, and the ac-tion of any given ideal polarization modulator can berepresented by a rotation (and�or an inversion) inthis space. Such an operation corresponds to the in-troduction of a phase delay between orthogonal po-larizations, which is the physical mechanism at work
D. T. Chuss ([email protected]), E. J. Wollack, and S. H.Moseley are with the NASA Goddard Space Flight Center, Code665, Greenbelt, Maryland 20771. G. Novak is with the Depart-ment of Physics and Astronomy, Northwestern University, 2145Sheridan Road, Evanston, Illinois 60208.
Received 11 October 2005; revised 5 January 2006; accepted 25January 2006; posted 31 January 2006 (Doc. ID 65313).
in a polarization modulator. The two degrees of free-dom of any given transformation are the magnitudeof the introduced phase delay and a parameter de-scribing the basis used to define the phase delay.These two parameters directly define the orientationand the magnitude of the rotation on the Poincarésphere: The rotation axis is defined by the spherediameter connecting the two polarization statesbetween which the phase is introduced, and themagnitude of the rotation is equal to that of the in-troduced phase.3
In order to measure the polarized part of a partiallypolarized signal, it is desirable to separate the polar-ized part of the signal from the unpolarized part. Thisis especially crucial when the fractional polarizationof the signal is small. One way to do this is to me-thodically change, or modulate, the polarized part ofthe signal (by changing one of the parameters of thepolarization modulator) while leaving the unpolar-ized part unaffected. Periodic transformations inPoincaré space can accomplish this encoding of thepolarized component of the signal for subsequent syn-chronous demodulation and detection. A convenientway of formulating the problem is to envision a de-tector that is sensitive to Stokes Q when projectedonto the sky in the absence of modulation. The polar-ization modulator then systematically changes thepolarization state to which the detector is sensitive.By measuring the output signal, the polarizationstate of the light can be completely characterized.
A common implementation of such a polarizationmodulator is a dielectric birefringent plate.4 A birefrin-gent plate consists of a piece of birefringent materialcut so as to delay one linear polarization componentrelative to the other by the desired amount (generallyeither to one half or one quarter of the wavelength ofinterest). In this case, the phase difference is fixed, andthe modulation is accomplished by physically rotatingthe birefringent plate (and hence the basis of the in-troduced phase).
In contrast, in this paper we explore a class of po-larization modulators in which the basis of phase in-troduction is held fixed, but the magnitude of the delayis variable. Throughout this paper, we will refer to anydevice that inserts an adjustable relative delay be-tween two orthogonal linear polarizations as a varible-delay polarization modulator (VPM). There are manyexamples of devices that implement a variable delayfor polarization modulation. For example, Martin5,and Martin and Puplett6 describe a version of theMartin–Puplett interferometer (MP) without an inputpolarizer. VPMs have also been used in several astro-physical polarimetry systems.7,8 In this work, we ex-plicitly separate the polarization modulation from thepolarized detection (analyzer) of the signal. Doing sohas two advantages. The first advantage is that thebasis of the VPM can be rotated at an arbitrary anglewith respect to the orientation of the analyzer. TheVPM in a MP is a specific example of this general casein which the relative angle of the VPM is 45° withrespect to the analyzer. It should be noted that Mar-tin5 considers misalignment errors between the beam
splitter and the analyzer in a MP. The analyticalexpression for the polarization in this case is thesame for the case of the single VPM placed at anarbitrary angle with respect to the analyzer, al-though the general physical implementation is differ-ent.
The development of the transfer function for a sin-gle VPM allows multiple copies of this device to becascaded at arbitrary relative orientations. We spe-cifically consider the case for two VPMs cascaded inseries at appropriate angles so as to fully sample allpossible Poincaré states. In this case, we are assum-ing a narrow enough passband such that the phasedelays introduced for the center wavelength approx-imately apply to the whole band. The VPMs are con-figured as follows: The VPM that is closest to thepolarization-sensitive detector has its beam-splittinggrid oriented at an angle of 45° with respect tothe axis of the detector (Q axis; see above), and theother VPM has its grid oriented at 22.5° with respectto the detector axes. We show how full modulation ofall linear and circular polarization states can beachieved with this device. The use of this architecturein a polarimeter that measures linear polarizationcan be understood as follows: If we set the deviceclosest to the source (VPM 1) for zero phase delay,and switch the VPM closest to the detector (VPM 2)between delays of 0 and �, then the detector axes, asprojected onto the plane of the sky, will switch be-tween Q and �Q. With VPM 1 set to a phase delay of�, switching VPM 2 between 0 and � will project thedetector axes to �U. The dual VPMs provide twodegrees of freedom, namely the phase delays of thetwo devices. The angles selected for the two basis setsare those for which the two degrees of freedom cor-respond to orthogonal coordinates on the Poincarésphere, thus allowing all polarization states to beaccessible to the detector.
There are several qualities that make this architec-ture a viable candidate technology for future astro-nomical polarimeters operating in the far infraredthrough millimeter regions of the spectrum. First,whereas a given birefringent plate can be built to mea-sure either circular or linear polarization, but not both,the VPM allows for implementations that cover theentire Poincaré sphere. Second, since the path differ-ence between orthogonal linear polarization states isvariable, these devices are easily retuned for use atmultiple wavelengths. Note also that since the VPMrequires no transmission through thick dielectric ma-terial, frequency-dependent antireflective coatings arenot required. Finally, this architecture requires onlysmall linear translations that will eliminate the needfor complicated systems of shafts, gears, and bearingsthat are common in birefringent plate modulators.9 Allof these qualities are beneficial to the future effort tomeasure the polarized flux of astronomical and cosmo-logical sources from space-borne telescopes.
In Section 2, we derive the Mueller matrix repre-sentation of the VPM using the interior part of aMP. Using this result, we calculate the frequency-dependent performance of a VPM in Section 3.
Section 4 describes an alternative architecture for theVPM that makes the dual modulator system feasible,Section 5 addresses possible systematics in the appli-cation of the VPM, and Section 6 describes laboratorytests to test polarization-modulating properties ofa single VPM. We conclude the paper with a briefsummary.
2. Single Variable-Delay Polarization Modulator
A Martin–Puplett interferometer consists of a VPMwith an analyzer on the output end nominally orientedat an angle of 45° with respect to the beam-splittinggrid. For spectrometer applications, one of the portson the input side is shorted by a grid oriented eitherparallel or perpendicular to the analyzer. In this sec-tion, we use the interior of the MP to generate theJones and Mueller matrices for a general VPM ori-ented at an arbitrary angle with respect to the opticalsystem in which it operates. A similar analysis has
been done by Martin,5 but the convenience of stan-dard polarization matrices allows for the subsequentgeneralization to the case of multiple VPMs in series.For the convenience of the reader, we have includedan appendix that briefly describes polarization ma-trix techniques and elucidates the mathematical con-ventions that we follow in our analysis.
A diagram of the VPM part of the MP is shown inFig. 1. Light enters from the left and is split into twoorthogonal polarizations by the 45° grid. The twocomponents of polarization are then sent to two roof-top mirrors that rotate the polarization by 90° withrespect to the grid wires. The beams recombine at thebeam splitter and exit the device at the top.
We will examine this device using Jones matrices,labeling the angle of the device to be the angle of thebeam-splitting grid as seen by the incoming radiation.We will first look at the case of a VPM at a rotation of45° and then generalize to an arbitrary angle using a
Fig. 1. The propagation of the electric field components and the �H, V� coordinate axes through the VPM part of a Martin–Puplettinterferometer at an angle of ��4 are shown. When d1 � d2, this device behaves like a mirror. When there is a path difference, it changesthe polarization state of the incoming radiation.
similarity transformation. For the simple case, theJones matrix representing this configuration,J� VPM���4�, can be expressed as the sum of the Jonesmatrices for the radiation in each of the armsof the VPM,
J� VPM��
4�� J� VPM�1���
4�� J� VPM�2���
4�. (2)
In turn, each of these terms can be decomposed intoa product of the Jones matrices of the individual el-ements in each optical path. The Jones matrices forthese elements are given in Table 2 in the Appendix.
J� VPM�1���
4�� J� WP��
4� J� z�d1� J� RT�0� J� z�d1� J� WP��
4�� � 1 1
�1 �1� exp�i4�d1���2 , (3)
J� VPM�2���
4�� J� WP���
4� J� z�d2� J� RT�0� J� z�d2� J� WT��
4�� �1 �1
1 �1� exp�i4�d2���2 . (4)
Making the definition � � 4��d2 � d1��� and setting� � ��2, we arrive at the following:
J� VPM��
4, ��� ½ei2��d1�d2���� cos � �i sin �
i sin � �cos � �. (5)
Next, we derive an expression for a VPM placed at anarbitrary angle . Recall that the definition of wehave chosen is the angle of the grid with respect to Hfor the radiation at the input port. To do this, wetransform into the coordinate system for which wehave already solved the problem, apply the transfor-mation for JVPM���4�, and then transform back. Inthe case of this architecture, there is a subtlety. Be-
cause the number of reflections is odd, the angle ofthe device as viewed from the outgoing light is thenegative of that viewed from the incoming light. Thisreflection is accounted for in Eq. (6). Setting �� � ��4�, we note that
In terms of the Stokes parameter basis set, thisexpression is
J� VPM�, �� � ei2��d1�d2��� �Q� cos � � i sin ��I� cos�2�� iV� sin�2�. (8)
Note that within a phase factor (which is irrele-vant in a measurement of power) JVPM � QJWP. Thismeans that the action of the VPM is equivalent tothat of a birefringent plate (having its birefringentaxis oriented at an angle � with a delay � � 2�)followed by a reflection (represented as the Jonesmatrix Q).
The matrix in Eq. (8) is unitary, and its deter-minant is �1. Thus its Mueller representation isexpected to describe symmetries on the Poincarésphere. By expanding the density matrices in thePauli matrix basis both before and after performingthe similarity transformation corresponding to theoptical system, one can generate the Mueller matrixfor the system,
This matrix can be expressed as a product of symme-try operations on the Poincaré sphere,
Here, �QV is a reflection at the Q–V plane, �QU is areflection at the Q–U plane, R� V is a rotation aroundthe V axis, and RQ is a rotation around the Q axis. Thematrix MWP�, �� is the Mueller matrix for a waveplate.10,11
J� VPM�, �� � R� †��� J� VPM��
4�R� �� �ei2��d1�d2���
2 �cos � � isin � sin 2 �isin � cos 2
isin � cos 2 �cos � � isin � sin 2�, (6)
J� VPM�, �� � ei2��d1�d2��� �cos � � i sin � cos 2 �i sin � cos 2
i sin � sin 2 �cos � � i sin � cos 2�. (7)
M� VPM�, �� ��1 0 0 00 cos2 2 � cos � sin2 2 �sin 2 cos 2�1 � cos �� sin 2 sin �
0 sin 2 cos 2�1 � cos �� �sin2 2 � cos � cos2 2 �cos 2 sin �
We assume that the detector at the back end of ouroptical system is sensitive to Stokes Q. This is essen-tially a statement about the orientation of the ana-lyzer in the polarimetric system. Strictly speaking, aQ sensitive detector requires a differencing of twoorthogonal linearly polarized detectors with an ori-entation that we choose to define the Q axis. How-ever, the following discussion also applies to the classof polarized detector strategies that collect only onelinear polarization. Such detectors are technicallysensitive to I � Q, but to lowest order or in idealmodulation, I does not couple to the polarization mod-ulation.
The modulator changes the polarization state ofthis detector as projected onto the sky. For a singleVPM, the polarization state that the detector mea-sured can be calculated from the second column of theMueller matrix,
Qdet � Qsky�cos2 2 � cos � sin2 2�� Usky sin 2 cos 2�1 � cos ��� Vsky�sin 2 sin ��. (11)
For VPM, is fixed and � is modulated. Using asingle VPM, it is not possible to completely modulateQ, U, and V. To see an example of this, consider thecase where is set to ��4. In this case,
Qdet � Qsky cos � � Vsky sin �. (12)
Here, Q and V are modulated, but U is not. This isbecause at this grid angle, U and �U propagatethrough the system separately without interfering.
That stated, the advantage of the VPM is that thephase freedom allows a straightforward method formodulating Q and V across large frequency bands. Webegin by defining � � k�, such that k � 2��� and � isthe total path difference between the two orthogonalpolarizations. [For the traditional Martin–Puplettbeam paths, � � 2�d2 � d1�.] The signal measured bya polarized detector would then be dependent on �,
Q��k, �� � Q�k� cos k� � V�k� sin k�. (13)
For bolometric detectors, the signal is integrated overthe instrument bandpass �k�,
Q���� � 0
�
Q��k�, �� �k��dk�
� 0
�
�Q�k��cos k�� � V�k��sin k�� �k��dk�.
(14)
Taking the Fourier transform of both sides yields
½�Q�k� � iV�k� ��� �1
2� �1
�2
Q���� eik�d�. (15)
It can be seen that the real part of the Fourier trans-form of the interferogram is the spectrum of Stokes Qfrom the source. The imaginary part is the Stokes Vspectrum. Note that broadband modulation relies onsampling a large enough range of path length differ-ences.
We note that for the standard implementation of aMartin–Puplett interferometer as a Fourier trans-form spectrometer, a horizontal or vertical grid isplaced at the input port of the device. This sets theinput polarization state to be purely Q, thus enabl-ing the internal grid to function as a broadband beamsplitter. This is equivalent to shorting one of the in-put ports. Thus if one makes the reasonable assump-tion that the polarization of the source is small, thenQ�k� �
12I�k�, and Eq. (15) reduces to the unpolarized
spectrum of the source. In this case, the device doesnot measure polarization, but relies on the fact thatthe input grid is setting the input polarization stateto something that is known.
The major disadvantage for this architectureis its insensitivity to Stokes U. For space-borneexperiments, U can be recovered by rotation ofthe spacecraft. For ground-based instruments, suf-ficient rotation is problematic, and thus an alterna-tive approach may be required.
An alternative to instrument rotation is to placetwo such devices in series. It is possible to calculatethe functional form of the polarization signal on thedetectors by simply chaining the two Mueller matri-ces together. The transfer equation of the optical sys-tem is now S� sky � M� VPM �1, �1� M� �2, �2� S� det. Notethat modulator 2 is closer to the detector than mod-ulator 1. Because our detectors are sensitive to onlyQ, we solve for the second column of the resultingmatrix,
Qdet � Qsky��cos2 21 � cos �1 sin2 21�� �cos2 22 � cos �2 sin2 22�� sin 21 cos 21 sin 22 cos 22�1 � cos �1�� �1 � cos �2� � sin 21 sin 22 sin �1 sin �2� Usky�sin 21 cos 21�1 � cos �1�� �cos2 22 � cos �2 sin2 22�� �sin2 21 � cos �1 cos2 21�sin 22 cos 22
� �1 � cos �2� � cos 21 sin 22 sin �1 sin �2� Vsky�sin 21 sin �1�cos2 22 � cos �2 sin2 22�� cos 21 sin 22 cos 22 sin �1�1 � cos �2�� sin 22 cos �1 sin �2. (16)
We now consider the specific case where 1 � ��8and 2 � ��4. These angles were carefully chosen toallow full sampling of the Poincaré sphere. Polarized
sensitivities for selected pairs of phase delay settingsfor the pair of modulators are shown in Table 1. Thesimplest method for modulating polarization is to as-sume a single phase delay over the entire bandwidth.In this case, one sets the modulators to the desireddetector polarization sensitivity and makes a mea-surement. One then repeats this measurement foreach state and builds up information about the po-larization state of the source.
One of the strengths of this modulator is its abilityto modulate quickly between different polarizationstates. This has the advantage of putting the polar-ization signal above the 1�f knee of the instrumentnoise spectrum. It may also be possible to extend thebandwidth in a way similar to that of the single mod-ulator above. One could scan these modulatorsthrough a range of delays and extract the frequency-dependent Stokes parameters.
4. Other Implementations
The architecture in Fig. 1 is not a unique implemen-tation of a VPM. In fact, there are several arrange-ments of grids and mirrors that correspond to Jonesmatrices that differ only by an absolute phase fromthose that describe the MP as derived above. The sim-plest of these designs is a system consisting of a polar-izing grid placed in front of a mirror. This design issimilar in structure to a reflecting wave plate,12 but inthis case, modulation occurs by modulating the grid-mirror spacing rather than by spinning the device.This alternative design for a polarizing interferometerhas been previously employed because of its compactfeatures and relative ease of construction.13 This im-plementation is useful for a dual modulator systemsince it requires significantly less space in the opticalpath than other implementations. The drawbacks aretwofold. First, the inability to achieve the zero pathlength condition causes a slight decrease in effectivebandwidth. Second, the two polarizations on the out-put side of the device are displaced slightly. This effectcan be mitigated by using the modulators at close tonormal incidence and by placing the modulators asclose as possible to a pupil.
5. Systematics
In developing a polarization modulator, one mustconsider the possibility of instrumental effects intro-duced by the action of the modulation. In a dielectrichalf-wave plate, such an effect arises from the ab-sorption properties of a birefringent material. Losstangents for light polarized along the fast and slowaxes are generally different. The result is a modu-lated signal that appears at twice the rotational fre-quency of the birefringent plate. For the dual VPMsystem, there are two important effects to consider.First, for different settings of the translational stage,the edge illumination will change, thereby potentiallyintroducing a spurious polarization signal. This prob-lem can be avoided by restricting the use of suchmodulators to slow optical systems in which the beamgrowth through the modulator is minimal. The sec-ond concern involves the differential absorption of thegrids and the mirrors of the modulator. For the roof-top mirrors, the incident angle of the radiation isthe same for different modulator positions. Thus theFresnel coefficients for each of the two polarizationswill remain essentially constant during the modula-tion process.
6. Experimental Results
A. Setup
To test the polarization modulation of a VPM, webuilt the MP interferometer configuration illustratedin Fig. 2. The beam exiting the horn attached to port1 is collimated by an ellipsoidal mirror. It then passesthrough a polarizing grid that has its wires orientedat 45° angle in projection. Each orthogonal polariza-tion is then launched down a separate arm of thedevice and reflects off a rooftop mirror that rotatesthe polarization vector by 90°. The beams recombineat the polarizer and are refocused into the feed con-nected to port 2. The setup is symmetric, and so thereverse light path is identical. The rooftop mirrorsare placed on translational stages such that theirrelative distances can be adjusted. The frequency-dependent phase that corresponds to this path lengthdifference is the parameter that determines the map-ping between polarization states on either end of thedevice. Note that this system can support �1 mode ineach polarization. Thus Gaussian analysis was usedin the design of the optics.14
From a microwave circuit perspective, this device isa four-port device with the two ports on either end ofthe device being defined by the vertically and horizon-tally polarized electric field modes. We use a HewlettPackard HP8106D millimeter wave vector networkanalyzer (VNA) to measure the scattering parametersbetween these modes. The calibration reference planeis shown (�1 and �2) in Fig. 2. The VNA can be used tomeasure the 2 � 2 scattering matrices of pairs ofthese ports, so to reduce contamination of our results,we place an orthomode transducer15 (OMT) at theback of each feed horn and terminate the unusedpolarization with a matched load. For the purposes ofthese measurements, it is useful to think of the end of
Table 1. Mapping of Qdet onto the Sky for Selected Values of �1 and�2 for Dual Modulators Having �1 � ��8 and �2 � ��4
the feed horn attached to port 1 of the VNA as thesource, and that attached to port 2 as the detector. Weset the polarization state of the source to be verticallypolarized light (a pure Q state) by orienting thewaveguide appropriately. On the detector side, wemeasure both V and H in successive measurementsby, respectively, omitting and adding a 90° twist tothe WR-10 waveguide between the OMT and the �2calibration point. We have measured the loss of thetwist to be 0.15 dB. The calibrated difference betweenthe power associated with H and V gives a measure-ment of Stokes Q at the detector.
The bandwidth of the test setup is approximately78–115 GHz. At the low end of the band, the bandedge is defined by that of the W-band feed horns, andat the high end, it is defined by the OMT return loss.
B. Experimental Procedure
We found the zero path length position by first max-imizing the signal in the V direction at a point wherethe S21 parameter was flat across the band. We thenwere able to use the first null condition to do a fineadjustment. V and H are measured for 27 combina-tions of positions of the two mirrors having pathdifferences corresponding to 24° steps in phase for� � 3 mm. Four sample spectra are shown in Fig. 3.We have included in these plots the expected trans-mission spectra [H � �1 � cos �� and V � �1 � cos ��],
adopting a global gain of 0.9 dB to account for theexpected loss beyond the calibration port. The returnloss of the system is about 26 dB and can be seen inthe H component of Fig. 3(a). The transmission effi-ciency of the horns is not constant across the bandand tends to roll off at low frequencies.
C. Results
This experimental setup is described mathematicallyby the expression in Eq. (12). In this case,
Qdet �H��� � f V���H��� � f V���
� Qsource cos � � Vsource sin �,
(17)
where � � 4��d2 � d1���. Here, H��� and V��� arethe powers corresponding to S21 when the twist isincluded and excluded, respectively. For each fre-quency, the relative gain factor, f, is calculated byfitting for the average values of the signals in the Hand V configurations and taking the ratio.
For each frequency, it is possible to measure StokesQ and V. The result of this analysis is shown in Fig.4. We find that the average Stokes parame-ters measured over the 78–115 GHz band areQ � �1.002 � 0.003 and V � 0.001 � 0.013. Thereis some nonzero power in Stokes V near the high end
Fig. 2. The Martin–Puplett in-terferometer is symmetrically fedby a pair of W-band feed horns(25–27 dBi) that are collimated byellipsoidal mirrors ( f � 25 cm).Each of the two rooftop mirrorsreflect a component of polariza-tions. The mirrors are mounted ontransports that are used to adjustthe path lengths of the individualpolarizations. The polarizing gridis mounted such that the wiresmake an angle of 45° with the rooflines in projection. The dashedand dotted lines show the posi-tions of the beam radius (–6.7 dBpower level) and –20 dB powerlevel, respectively, of a Gaussianbeam propagating through thestructure for a 26 dBi feed and awavelength of 3 mm (100 GHz).We have illustrated the location ofthe 90° twist on port 2 that con-verts the sensitivity of port 2 fromV to H. The calibration referenceplane is also shown (�1 and �2).
of the band. It is unclear as to whether this is due toa systematic effect or due to an unknown source po-larization.
D. Resonances
In this laboratory setup, proper termination of the un-used port at both the entrance and exit apertures isessential, as even small reflections can introduce res-onances. These resonances are an indication of thelevel of uncertainty of phase control of the radiationpropagating through the device. This uncertainty di-rectly leads to a frequency-dependent random mixingbetween the Q and V polarization states and hence adecrease in the precision of the MP as a polarimeter.We have found that this systematic “noise” can becontrolled by various levels of termination of the un-used polarization. The addition of the OMTs in thesignal chain reduced the noise in the transmissionfrom 3 to 1 dB. We also added a horizontal grid at themouth of the source feedhorn to redirect any residualH component to an eccosorb beam dump. This gridreduced the noise in transmission to 0.25 dB and alsoreduced the average coupling of Q into V from 4% tounder 1%.
On a telescope, this problem is mollified by the fact
that the source port is nearly perfectly terminated onthe sky. This greatly reduces phase uncertaintiesin the system as well as the need for excessive polar-ization filtering.
7. Summary
We have described a technique for polarization modu-lation in which n phase delays between linear orthog-onal polarizations are placed in series with arbitraryrelative orientations. We specifically consider then � 1 and n � 2 cases, and find that for appropriaterelative orientations, it is possible to fully modulatethe polarization in the n � 2 case. In the far infraredthrough submillimeter, where bandpasses are typi-cally ���� � 0.1, this device can be used in a similarmanner to a half-wave plate. Broader bandpasses����� � 0.3� may be accommodated by using morecomplex modulation schemes. This architecture en-ables one to construct a modulator that can be maderobust, broadband, and easily tunable to differentwavelengths. In addition, it allows for the completedetermination of the polarization state of the incom-ing radiation by the measurement of Stokes Q, U,and V.
It is worth noting one final potential application of
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Fig. 3. The transmission spectra are shown for four different values of d1 � d2: (upper left) �13 �m, (upper right) 587 �m, (lower left)�813 �m, and (lower right) �1013 �m. The thick solid line in each plot is the spectrum of the vertical (V) linear polarization measuredat port 2 of the VNA. The thin solid line in each plot is the spectrum of the horizontal (H) linear polarization measured at port 2. The Hpolarization is measured by adding a 90° twist in the WR-10 waveguide attached to port 2 of the VNA. Theoretical predictions for H andV are plotted as thick and thin dashed lines, respectively.
VPMs. The ability of these devices to work at roomtemperature may make them useful as calibrators.An input polarized signal can be transformed quiteeasily to test the polarization response of a precisionpolarization sensor. It can transform an initially lin-early polarized state into an elliptical polarizationstate.
Appendix A: Polarization Matrix Methods
In this appendix, we review the properties of Jones,density, and Mueller matrices and their relation-ships.
A. Jones Matrices
Jones matrices16 are a convenient way to analyzeradiation as it propagates through an optical systemin architectures in which it is important to keep trackof phase. For the ideal case, we assume that all portsare matched and so no cavities are formed. This for-mulation is applicable for coherent radiation; how-ever, it can be extended using the closely relatedformalism of density matrices to treat the problem ofpartially polarized light.17
Jones matrices are 2 � 2 matrices that containinformation about how the orthogonal electric fieldcomponents transform in an optical system. The in-put Jones vector is defined as follows:
�E � � �Ex
Ey�� �EH
EV�. (A1)
The output vector from an optical system can then berepresented by |Ef� � J� |Ei� where J� is the vectortransformation introduced by the optical system. Thepower measured at a detector at the back end of sucha system is given by
�Ef�J� det�Ef � � �Ei�J�† J� det J� �Ei � . (A2)
The matrix J� det is dependent on the properties of thedetector used to make the measurement.
In the Jones matrix representation, Stokes param-eters are represented by the Pauli matrices and theidentity matrix.
I� � �0 � �1 00 1�, Q� � �1 � �1 0
0 �1�,U� � �2 � �0 1
1 0�, V� � �3 � �0 �ii 0 �. (A3)
We will use the convention that a bar over theStokes symbol indicates its Jones matrix representa-tion. An unbarred Stokes parameter represents mea-surable power (e.g., Q � �E�Q� �E�). Note that themeasured power in each of the Stokes parameters is
I � �E|I�|E � � EH2 � EV
2, (A4)
Q � �E|Q� |E � � EH2 � EV
2, (A5)
U � �E|U� |E � � 2 ��EH*EV�, (A6)
V � �E|V� |E � � 2 ��EH*EV�. (A7)
These equations connect the Jones matrix formula-tions of the Stokes parameters to their familiardefinitions.3,18 These four Stokes matrices havethe following multiplicative properties. Defining��0, �1, �2, �3) � �I�, Q� , U� , V� �, �0�� � ���0� �� for� � �0, 1, 2, 3� and �j�k � �l �jkli�l � �jk�0 forj, k, l � �1, 2, 3�. These four matrices form a conve-nient basis for expressing Jones matrices. Table 2shows both the explicit Jones matrices and the Stokesexpansion for selected optical transformations. Themirror transformation, which can be expressed sim-ply as Q, sets the convention for how the �H, V� coor-dinate system is propagated through the opticalsystem. Note that for some structures, the Stokesexpansion provides a convenient way to express op-tical elements. Successive transformations can be cal-culated either by matrix algebra or by the Paulialgebra defined above.
B. Density Matrices
In the general case of partially polarized light, polar-ization arises because of time-averaged (statistical)correlations between the electric field components.The density matrix is a complex 2 � 2 matrix thatfully characterizes the polarization state of the light.It is given by
D� � ��Ex* Ex� �Ex* Ey��Ey* Ex� �Ey* Ey��. (A8)
Here, the brackets indicate a time average. If thedensity matrix is expressed as a linear combination of
Fig. 4. The normalized Stokes parameters q and v are calculatedas a function of frequency by fitting to the 27 mirror positions. Themean values of q and v across the 78–115 GHz band are �1.002 �0.003 and 0.001 � 0.013, respectively.
the Pauli matrices, D� � I�0 � Q�1 � U�2 � V�3, thecoefficients are the Stokes parameters.
The transformation of the polarization state by anoptical system is given by a similarity transforma-tion, D� � � J� †D� J� . Here, J is the Jones matrix describ-ing the optical system. For the purposes of this paper,we are interested in how the polarization state of thedetectors map onto the sky, and so D� sky � J� †D� detJ� .Note the similarity in the transformation of the den-sity matrix and the expression for total power in theJones matrix formalism [Eq. (A2)].
C. Mueller Matrices
Until now, no limitations have been placed on J, thematrix describing an optical system under consider-ation. If the magnitude of the determinant of J isunity, then there is a homomorphism between thegroup of 2 � 2 matrices having |det�J� �| � 1 and thePoincaré or inhomogenous Lorentz group. In thiscase, the quantity I2 � Q2 � U2 � V2 is preservedunder these transformations. In analogy to specialrelativity,20 the inhomogeneous Lorentz group can berepresented by a group of 4 � 4 real matrices actingon a Stokes vector, S � �I, Q, U, V�. These matricesare known as Mueller matrices. For our purposes, weconsider the Mueller matrix that maps the Stokesparameters at the detector to the sky: S� sky � M� S� det.
For polarization modulation, we are particularly in-terested in the case for which the Jones matrices de-scribing our optical system are unitary. In this case,Stokes I decouples from the other Stokes parametersand the quantity P 2 � Q2 � U2 � V2 is preserved.This subgroup can be represented by 3 � 3 orthogo-
nal submatrices that represent symmetries on thesurface of a sphere in a space having Stokes Q, U, andV as axes. This sphere is called the Poincaré sphere.
As an aside we note that if we restrict the group ofdensity matrices to those with positive determinants,the system is described by SU(2), and thus there is ahomomorphism between this group and SO(3), thegroup of rotations on the Poincaré sphere. These arethe groups that are important to a wave plate; how-ever, the physical reflection involved in the VPMarchitecture introduces a negative determinant, re-sulting in combinations of rotations and reflections onthe Poincaré sphere.
The authors thank Don Jennings for his usefuldiscussions on polarimetry; Dale Fixsen for his helpwith the manufacturing of the collimators; TerryDoiron for supplying the grids; and Gary Hinshaw,Dominic Benford, and Al Kogut for their support ofthis work. This work was funded by a NASA GSFCDDF award and by NASA ROSS APRA grantAPRA04-007-0150.
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Description Symbol Matrix Representation Stokes Expansion
Linear distance J� z�d� �exp�i2�d��� 00 exp�i2�d���� I� exp�i2�d���
Mirror J� M �1 00 �1� Q�
Wire grid (ref.) J� WP�� � cos2 sin cos
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Wire grid (trans.) J� WT�� � sin2 �sin cos
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Coord. rotation R� �� � cos sin
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