A quantum mechanical approach to establishing the magnetic field orientation from a maser Zeeman profile
J. A. Green, M. D. Gray, T. Robishaw, J. L. Caswell, N. M. McClure-Griffiths
MMon. Not. R. Astron. Soc. , 1–10 (XXXX) Printed 20 October 2018 (MN L A TEX style file v2.2)
A quantum mechanical approach to establishing themagnetic field orientation from a maser Zeeman profile
J. A. Green (cid:63) , , M. D. Gray , T. Robishaw , J. L. Caswell ,and N. M. McClure-Griffiths SKA Organisation, Jodrell Bank Observatory, Lower Withington, Macclesfield, SK11 9DL, UK CSIRO Astronomy and Space Science, Australia Telescope National Facility, PO Box 76, Epping, NSW 1710, Australia Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, University of Manchester,Manchester, M13 9PL, UK National Research Council Canada, Herzberg Astronomy and Astrophysics Programs, Dominion Radio Astrophysical Observatory,PO Box 248, Penticton, BC V2A 6J9, Canada
Accepted 2014 February 28. Received 2014 February 13; in original form 2013 August 7
ABSTRACT
Recent comparisons of magnetic field directions derived from maser Zeeman split-ting with those derived from continuum source rotation measures have prompted newanalysis of the propagation of the Zeeman split components, and the inferred fieldorientation. In order to do this, we first review differing electric field polarization con-ventions used in past studies. With these clearly and consistently defined, we then showthat for a given Zeeman splitting spectrum, the magnetic field direction is fully deter-mined and predictable on theoretical grounds: when a magnetic field is oriented awayfrom the observer, the left-hand circular polarization is observed at higher frequencyand the right-hand polarization at lower frequency. This is consistent with classicalLorentzian derivations. The consequent interpretation of recent measurements thenraises the possibility of a reversal between the large-scale field (traced by rotationmeasures) and the small-scale field (traced by maser Zeeman splitting).
Key words: masers – polarization – magnetic fields – radiative transfer
Although a large number of magnetic field studies have beenundertaken using Zeeman splitting of maser spectra (e.g.Fish et al. 2005; Surcis et al. 2011), the majority of thesestudies only consider magnetic fields for individual regions.For mapping the field pattern within a source, the intensityof the field is of prime interest, together with changes infield direction, but knowledge of the actual line-of-sight fieldorientation (either towards or away from the observer) is notusually of importance to the interpretation.However, when considering ensembles of sources, thereis a possibility of comparing absolute field directions withGalactic structure, and with measurements obtained byother techniques. Results of the MAGMO survey (Greenet al. 2012), and prior observations of magnetic field orienta-tion from hydroxyl (OH) maser Zeeman splitting (e.g. Reid& Silverstein 1990; Fish et al. 2003; Han & Zhang 2007),have led us to re-evaluate the field direction for a given Zee-man pattern. Specifically, we address the apparent contra-diction in field direction between the maser measurements (cid:63)
E-mail:[email protected] and those inferred from Faraday rotation (e.g. Brown et al.2007; Van Eck et al. 2011) by exploring the Zeeman splittingin the quantum mechanical sense.In the weak field limit, Zeeman splitting causes the oth-erwise degenerate energy levels of an atom or molecule tosplit into 2 J + 1 magnetic components, where J is the totalangular momentum quantum number. In the simplest caseof a J = 1 − : the unshifted (in frequency relative to zeromagnetic field) π and the two shifted σ s, denoted σ + and σ − (Figure 1). Commonly, conventions are invoked when at-tributing the σ + and σ − components to a handedness ofcircular polarization, and for allocating which of these isfound at the higher frequency for a given field direction.In this paper we first outline the current convention forinferring field orientation from an observed maser spectrum We focus on this simple instance, applicable to the OH doublettransitions at 1665 and 1667 MHz, and H i at 1420 MHz. We notethat similar analysis can be applied to the more complex Zeemanpatterns of some other transitions, such as the 1720 MHz satellitetransition of OH.c (cid:13) XXXX RAS a r X i v : . [ a s t r o - ph . GA ] A p r Green et al. ! !"" %&%%'%(&)% *%%%%% % ! +,- %&%%'%(&)% % " . ! /0&% . ! /(&%. ! /'% %%%%%%% % Figure 1.
Transitions between magnetic sub-levels of Zeemansplitting. ∆ m = m lower − m upper (e.g. Garcia-Barreto et al. 1988;Gray & Field 1994; Gray 2012). ∆ m = +1 has the lower frequency(higher equivalent Doppler radial velocity), ∆ m = − (Section 2). We then re-evaluate the propagation of the in-dividual components to show how the field direction is fullydetermined and predictable on theoretical grounds, and isconsistent with the previously used convention (for exampleas adopted in Davies 1974 and Garcia-Barreto et al. 1988).The argument is presented first in an abbreviated descrip-tive form (Section 3) before a full derivation (Section 4). Fur-thermore, in the appendix we test the compliance of variousmaser theory publications that discuss polarization (consid-ering the direction of waves, the standard Cartesian axissystem and the polarization conventions). Conventions of field orientations have a long and chequeredpast, exacerbated by differences between optical and radiowavelengths, emission and absorption, the pulsar commu-nity and the rest of the astronomy community (e.g. Babcock& Cowling 1953; Babcock 1953). The use of polarizationconventions in theoretical papers over the years has simi-larly been inconsistent. The handedness of polarization intheoretical work is determined by the pair of helical vectors(in the spherical coordinate basis) that are used to repre-sent left-hand circular polarization (LCP) and right-handcircular polarization (RCP). A detailed history of the earlymeasurements and conflicting conventions is given by Ro-bishaw (2008).There are essentially three elements that have to betaken into account to define the field direction: 1. what isdefined as RCP and LCP polarization (invoking coordinatesystems and basis vectors); 2. which σ components thesepolarizations interact with; and 3. which frequencies thesepolarizations are found at for a field towards us or away fromus. The IEEE convention is the current standard for thefirst element, defining LCP as clockwise rotation of the elec-tric field vector as viewed by the observer with radiation ap-proaching, and RCP as counterclockwise (see also Figure 2).Radio astronomers adopted the IEEE usage, and it was for-mally endorsed in 1973 by the IAU (Commission 40 chairedby G. Westerhout). Unfortunately, an opposite widely usedconvention is adopted in classical optics, by both physicistsand optical astronomers. Tested sets of helical vectors inlater sections may therefore be described as either IEEE-compliant or optics-compliant.For the next two elements, we consider both observa-tions and the IEEE convention for Stokes V . The definitionof Stokes V is required for field directionality as discussedlater. The IAU convention is that: Stokes V is RCP minusLCP, therefore RCP corresponds to positive V and LCP tonegative V , i.e. V = (RCP − LCP) = ( ˜ E R ˜ E ∗ R − ˜ E L ˜ E ∗ L ) , (1)the second expression being the representation in terms ofelectric field amplitudes of the two polarizations as helicalvectors in the spherical basis (e.g. Landau et al. 1982). Thetilde indicates a complex-valued function and the asteriskthe complex conjugate.In order to apply these conventions to observations, it isalso necessary to know whether an observed shift of LCP tolower frequency, i.e. equivalent higher Doppler radial veloc-ity (and RCP to higher frequency, or lower velocity) corre-sponds to a field oriented towards or away from the observer.An early paper where this is an issue of special interest isDavies (1974), where the field direction for a group of sourcesis compared to the direction of Galactic rotation. That paperasserts that RCP shifted to higher velocity (as in the caseof the much studied W3(OH) region) corresponds to a fieldaway from the observer. The paper also describes this fieldorientation as a positive magnetic field. These are the sameconventions used in earlier papers considering H i absorption(Davies et al. 1962; Verschuur 1969). All subsequent papersthat we are aware of, and in particular the commonly citedpaper by Garcia-Barreto et al. (1988) have also retainedthis convention. None of the papers show a derivation jus-tifying this convention, and the later papers in particularhave merely adopted the convention without reassessing ifit is correct.However, accepting the above assertion, or conven-tion, the magnetic field orientation from Zeeman splittingof maser emission for the Carina-Sagittarius spiral arm isfound to be opposite to that indicated by rotation measuresof Galactic and extragalactic sources (Green et al. 2012,and references therein). It is this apparent discrepancy thatprompted a rigorous re-evaluation. The radio engineering definition of RCP and LCP dates from1942 (as decreed by the IRE, Institute of Radio Engineers) butis commonly referred to as the IEEE standard (endorsed in 1969by the IEEE, which had been formed in 1963 as a merger of IREand IEE). We note that in this paper, confusingly, the labelled Stokes V has a sign inconsistent with the IAU definition of V .c (cid:13) XXXX RAS, MNRAS , 1–10
M approach to Zeeman magnetic field orientation y xDirection of rotation forIEEE rhc−polarization NorthEast z Figure 2.
The ‘right-handed’ axis system with the electric field ofthe wave from equation (4) sketched in, and showing the directionof rotation of the electric field vector, as seen by the observer, ofRCP radiation under the IEEE convention. The alignment of the x and y axes with, respectively, North and astronomical Eastfollows the standard IAU orientations as set out in (Hamaker &Bregman 1996). Although there has been much ambiguity within the astro-nomical community, there is a significant body of physicsliterature with derivations, in a classical sense, of the field di-rection from Zeeman splitting. These started with the origi-nal work by Zeeman (1897, 1913) and include: White (1934);Sommerfeld (1954); Stone (1963); Jenkins & White (1976);Landi Degl’Innocenti & Landolfi (2004); Haken et al. (2005).In this work if the magnetic field is directed towards the ob-server (and denoted with a negative value), IEEE LCP is atthe lower frequency, IEEE RCP is at the higher frequency.Similarly if the magnetic field is directed away from the ob-server (and denoted with a positive value) IEEE LCP isat the higher frequency, IEEE RCP is at lower frequency.Throughout the rest of the paper we refer to this body ofwork as the ‘classical Lorentzian derivation’.
In this section, we revisit the quantum mechanics and radia-tive transfer of the Zeeman effect to demonstrate that theinferred direction of the field is uniquely defined by the ob-served frequency (or velocity) shift, in accordance with theclassical Lorentzian derivation. σ components We now consider the interaction of circularly polarized ra-diation with molecules that are subject to Zeeman split-ting by an external magnetic field (Figure 3). Accordingto the discussion in Eisberg & Resnick (1974), the mag-netic moment of the molecule, µ , is close to, but not ex-actly, anti-parallel to the total angular momentum vector, J , (or F in a molecule like OH or CH that has a Zeemaneffect of hyperfine structure). The magnetic moment pre-cesses rapidly about − J and much more slowly about − B .Eisberg & Resnick (1974) introduce the approximation that in one period of rotation of µ about − B , µ will rotate somany times about − J that the component of µ perpendic-ular to − J averages out to zero, and we need to consideronly the parallel component, µ J , precessing with − J about − B . This precession implies a corresponding precession of J about B , and it may be shown (for example Littlefield& Thorley 1979) that the sense of this latter precession iscounterclockwise for observers with the magnetic field point-ing towards them. Aligning the magnetic field and radiationpropagation directions along the z -axis ( θ = 0, see Figure 3),observers receiving the radiation also see J rotating counter-clockwise, corresponding to right-handed rotation under theIEEE convention (Figure 4). Since m is the quantum num-ber corresponding to J z , the projection of J on the z -axis,this right-handed (counterclockwise) rotation corresponds topositive values of m .The considerations above allow us to consider theradiation-molecule interaction in a σ + transition (Figure 3).Recall that in our convention, a σ + transition is one in whichthe value of m increases by 1 in emission. The left-handside of the figure shows the transition in absorption. Themolecules change from the initial state (a) as discussed above(IEEE right-hand rotation and m = 1) to the final state (b)where J has no projection on the z -axis and m = 0. Theoverall value of m must therefore be zero. To conserve theangular momentum of the interaction, the initially right-handed molecules must interact with LCP radiation (underthe IEEE convention), which has an electric field vector thatrotates clockwise as viewed by an observer receiving the ra-diation (Figure 3(a)). A derived result of this scheme is thata photon of the LCP radiation must carry –1 unit of angu-lar momentum associated with m . This result is consistentwith the conventions on photon polarization in Landau et al.(1982) and Fujia Yang (2010), having taken into account thehandedness conventions used in these works. On the right-hand side of Figure 3 we see a stimulated emission event,with the LCP radiation now approaching a molecule with m = 0 (part (c)). A photon of this radiation then copiesitself, and leaves the molecules in right-hand precession (d).The overall value of m in this stimulated emission case is −
1. It should be noted that for the case of spontaneousemission, one can follow the left side of Figure 3 from (b)to (a): start with m = 0 and no radiation and the resultis radiation with LCP, and the molecules have undergonea right-handed transition, increasing m by 1; this is in thesame sense as for stimulated emission.For Zeeman splitting of maser emission, if ∆ m = m lower − m upper (e.g. Gray & Field 1994; Gray 2012), where m lower and m upper are the quantum numbers correspond-ing to the magnetic sub-levels (Figure 1), we find that σ + is always found at the lower frequency (higher velocity), σ − at the higher frequency (lower velocity). This is shownschematically in Figure 1. This convention is presented in the often cited Garcia-Barretoet al. (1988), although it should be noted that the alternative∆ m = m lower − m upper is also often adopted (as noted byD. E. Rees in Kalkofen 1988).c (cid:13) XXXX RAS, MNRAS , 1–10
Green et al.
Absorption Stimulated Emission (a) (b) (d) (c) z z z z B B B B J J x x x
Figure 3.
Radiation (green dashed lines and solid circles) andmolecule (black solid lines and dashed circles) interaction in a σ + transition. The left side represents the interaction for absorptionof radiation, with an IEEE right-hand rotating molecule absorb-ing LCP radiation, the right side represents the interaction forstimulated emission, with incident LCP radiation on an unpolar-ized molecule resulting in a IEEE right-hand rotating moleculeand twice the LCP radiation. (a) and (c) show the initial statesof the interaction, (b) and (d) the final states. Radiation is prop-agating in the + z direction. !" dVdz = ! (1 + cos " ) V + ! cos " I !" !" !$ !" !$ (Gray & Field 1994) >&%8'*':%+ ! B dVdz = ! (1 + cos " ) V ! ! cos " I !"0 !$0 CDD>
Figure 4.
Definition of propagation and field direction vectors, θ represents the angle between the two, and the observer is lookingtowards the z -axis from below. σ to Stokes V correspondence The evolution of Stokes V with propagation distance forthe two σ components, following Goldreich et al. (1973),hereafter GKK73, can be defined respectively for the σ + and σ − components as : dV /dz = γ (1 + cos θ ) V − γcosθI (2) dV /dz = γ (1 + cos θ ) V + 2 γcosθI, (3)where γ is the gain coefficient, and with the field vectorsand θ defined as in Figures 3 and 4. Remembering thatStokes I exceeds Stokes V , it can be seen from this equationthat for an aligned field, one where the magnetic field vec-tor is approximately coincident with the propagation vector Note that the ± σ ± notation used in the present work. ( θ ≈ σ + com-ponent will have increasingly negative Stokes V and the σ − component increasingly positive Stokes V . Similarly for anopposing field direction ( θ ≈ π ), the σ + component willhave increasingly positive Stokes V and the σ − componentincreasingly negative Stokes V . Thus, with the frequencies(or equivalent velocities) of the σ + and σ − components de-fined by the quantum mechanics of the splitting (Section3.1), we know inherently that positive Stokes V at a lowerfrequency (higher velocity) indicates a field directed awayfrom the observer. If we now, as is commonly done, invoke the IAU definitionof Stokes V for emission, with positive V corresponding toRCP and negative V corresponding to LCP, we see thatRCP at a lower frequency (higher velocity) and LCP at ahigher frequency (lower velocity) indicates a field away fromthe observer. In this section we justify sections 3.2 and 3.3 by demonstrat-ing the consistency of the statement based on helical basisvectors.
We consider our electromagnetic (EM) waves to propagatein the positive z direction. The electric and magnetic fieldsare then confined to the xy -plane (Figure 2). The electricfield of such a plane-polarized wave has a standard repre-sentation (Young & Freedman 2004; Lothian 1957; Jenkins& White 1957) of: E ( z, t ) = ˆ x E x cos( ωt − kz ) , (4)where ˆ x is the unit vector along the x -axis, E x is the fieldamplitude, ω is its angular frequency and k = ω/c , itswavenumber. In equation (4), the field amplitude is assumed to be realand constant. In maser astrophysics, we typically deal witha spectral line composed of Fourier components distributedabout a line centre frequency, ω . The width of the line isnarrow in the sense that some width parameter, ∆ ω , such asthe full width at half maximum, satisfies ∆ ω (cid:28) ω . We cannow generalise the field in equation (4) to the typical masercase by letting the rapidly oscillating trigonometric termdepend on ω , and by introducing a slowly-varying phasefactor into the amplitude of each Fourier component: see forexample, Menegozzi & Lamb (1978); Goldreich et al. (1973);Deguchi & Watson (1990); Gray (2012). The complex ampli-tude of the full field, ˜ E x ( z, t ) then becomes an integral overfrequency offset from ω , and varies on a timescale vastlylonger than 1 /ω . The field is now, E ( z, t ) = (cid:60) (cid:8) ˆ x ˜ E x ( z, t ) e − iω ( t − z/c ) (cid:9) , (5) c (cid:13) XXXX RAS, MNRAS000
We consider our electromagnetic (EM) waves to propagatein the positive z direction. The electric and magnetic fieldsare then confined to the xy -plane (Figure 2). The electricfield of such a plane-polarized wave has a standard repre-sentation (Young & Freedman 2004; Lothian 1957; Jenkins& White 1957) of: E ( z, t ) = ˆ x E x cos( ωt − kz ) , (4)where ˆ x is the unit vector along the x -axis, E x is the fieldamplitude, ω is its angular frequency and k = ω/c , itswavenumber. In equation (4), the field amplitude is assumed to be realand constant. In maser astrophysics, we typically deal witha spectral line composed of Fourier components distributedabout a line centre frequency, ω . The width of the line isnarrow in the sense that some width parameter, ∆ ω , such asthe full width at half maximum, satisfies ∆ ω (cid:28) ω . We cannow generalise the field in equation (4) to the typical masercase by letting the rapidly oscillating trigonometric termdepend on ω , and by introducing a slowly-varying phasefactor into the amplitude of each Fourier component: see forexample, Menegozzi & Lamb (1978); Goldreich et al. (1973);Deguchi & Watson (1990); Gray (2012). The complex ampli-tude of the full field, ˜ E x ( z, t ) then becomes an integral overfrequency offset from ω , and varies on a timescale vastlylonger than 1 /ω . The field is now, E ( z, t ) = (cid:60) (cid:8) ˆ x ˜ E x ( z, t ) e − iω ( t − z/c ) (cid:9) , (5) c (cid:13) XXXX RAS, MNRAS000 , 1–10
M approach to Zeeman magnetic field orientation where the tilde on the amplitude indicates a complex-valuedfunction. As discussed above, the wave in equation (5) ismoving in the direction of more positive z .For the present purpose of testing the conformity of EMradiation definitions with polarization conventions, the fullcomplexity of equation (5) is not required. Our investiga-tions require the use of time differences of the order of 1 /ω and distances vastly shorter than any amplification or gainlength. We therefore ignore the slow time and space depen-dence of the complex amplitude, leaving it in the form ofa constant real amplitude multiplied by a constant phasefactor, e iφ x , that is: E ( z, t ) = (cid:60) (cid:8) ˆ x E x e iφ x e − iω ( t − z/c ) (cid:9) . (6) The wave represented by equation (6) is linearly polarized inthe xz -plane. In this work we need to consider circularly and,more generally, elliptically polarized radiation. This will con-tain both x and y components of the electric field, each withits own phase factor. In general, we have: E ( z, t ) = (cid:60) (cid:8)(cid:2) ˆ x E x e iφ x + ˆ y E y e iφ y (cid:3) e − iω ( t − z/c ) (cid:9) , (7)where φ x and φ y are the phases of the Cartesian field com-ponents.A Cartesian representation is unwieldy when calculat-ing the interaction of the the EM radiation with the molecu-lar density matrix, so it is customary to shift to a set of unitvectors based on positive and negative helicity: helical unitvectors in the spherical basis, often written as ˆ e + and ˆ e − .The field amplitude can now be broken into helical, ratherthan Cartesian, components, so that the usual representa-tion for our elliptically polarized wave is: E ( z, t ) = (cid:60) (cid:8)(cid:2) ˆ e + ˜ E + ( z, t ) + ˆ e − ˜ E − ( z, t ) (cid:3) e − iω ( t − z/c ) (cid:9) . (8)There is a third helical unit vector, but this is simply equal tothe z -axis unit vector, and is often written, ˆ e = ˆ z . The posi-tive and negative helicity unit vectors are unfortunately am-biguous, and we discuss below how to attach them to a stan-dard pair of unit vectors corresponding to IEEE LCP andRCP. We have removed one ambiguity by choosing to write e − iω ( t − z/c ) (rather than e + iω ( t − z/c ) ) when using complexexponential notation, as appears to be standard practice inmaser polarization theory papers, including GKK73. We have defined our EM wave to propagate along the z -axis in the positive direction in Section 4.1. This definitionmust be supplemented by a convention for the orientationof the x and y axes if any test of handedness is to work.We assume that the standard ‘right-handed’ system of axesfrom mathematics has been used by all authors unless theyhave clearly stated otherwise. This axis system is drawn inmany textbooks, and it is reproduced in Figure 2 (Arfken1970; Boas 1966). Also shown in Figure 2 is the EM wavefrom equation (4) at time t = 0. In mathematical descriptions of elliptically polarized radia-tion, it is the helical unit vectors that decide the handednessof polarization, given some standard definition of left andright. Here, we present a formal prescription for testing anypair of helical unit vectors against the IEEE standard:(i) Associate the positive and negative helicity unit vec-tors with presumed LCP and RCP radiation.(ii) Use equation (8) with the presumed RCP and LCPunit vectors to determine the RCP and LCP electric fieldcomponents in terms of their Cartesian counterparts.(iii) Write down a version of equation (8) correspondingto an RCP wave.(iv) Insert into this equation the definition of the pre-sumed right-hand unit vector, and resolve the electric fieldinto its Cartesian components.(v) Set a fixed distance, say z = 0.(vi) Pick a time, t , such that the electric field is alignedalong the positive y -axis.(vii) Advance the time to t so that ω t = ω t + π/ t is aligned with the negative x -axis, then the field has rotated counterclockwise from theobserver’s point of view (see Figure 2) and the presumedRCP vector conforms to the IEEE standard. If the field hasinstead rotated clockwise from the observer’s point of viewto point along the positive x -axis, then the presumed right-handed unit vector is actually left under the IEEE conven-tion (or right-handed in the optics convention). The seminal theory paper on maser polarization is GKK73.Their equation 12 clearly defines Stokes V in accordancewith the IAU convention: right minus left. The same equa-tion also tells us that the positive helicity unit vector shouldbe associated with RCP (and negative helicity with LCP).The definitions of the helical unit vectors are given in thetext just above equation 12 of GKK73 and, given the asso-ciations above, we deduce that,ˆ e R = (ˆ x + i ˆ y ) / √ e L = (ˆ x − i ˆ y ) / √ , (9)noting that in this basis the left-hand vector is the complexconjugate of the right-hand vector. Do these vectors conformto the IEEE definition? The short answer is yes, they do. Toprove this, we follow the prescription set out in Section 4.5above.Step (i) of Section 4.5 has already been completed inthe discussion above. For step (ii), we write down a versionof equation (8) in which the positive and negative helicityvectors and components are replaced by their RCP and LCPequivalents: E ( z, t ) = (cid:60) (cid:8)(cid:2) ˆ e R ˜ E R + ˆ e L ˜ E L (cid:3) e − iY (cid:9) , (10)where Y ( z, t ) = ω ( t − z/c ). Inserting the definitions fromequation (9), and resolving into Cartesian components, weobtain, E ( z, t ) = (1 / √ (cid:60) (cid:8)(cid:2) ˆ x ( ˜ E R + ˜ E L ) + i ˆ y ( ˜ E R − ˜ E L ) (cid:3) e − iY (cid:9) , (11) c (cid:13) XXXX RAS, MNRAS , 1–10
Green et al. from which it is evident that E x = ( ˜ E R + ˜ E L ) / √ E y = i ( ˜ E R − ˜ E L ) / √
2. Inverting this pair of expressions requiresthat,˜ E L = ( E x + i E y ) / √ E R = ( E x − i E y ) / √ . (12)We can now complete step (iii) by writing down a version ofequation (8) in (presumed) RCP only: E R ( z, t ) = (cid:60) (cid:8) ˆ e R ˜ E R e − iY (cid:9) , (13)and continue to step (iv) by inserting the definitions of ˆ e R from equation (9) and of ˜ E R from equation (12). The resultis, E R ( z, t ) = (1 / (cid:60) { [ˆ x E x + ˆ y E y + i (ˆ y E x − ˆ x E y )] × (cos Y − i sin Y ) } . (14)Step (iv) is completed by multiplying out the brackets andtaking the real part, leaving E R ( z, t ) = (1 / { ˆ x ( E x cos Y − E y sin Y )+ ˆ y ( E y cos Y + E x sin Y ) } . (15)For step (v), we set z = 0, so that Y (0 , t ) = ω t in equa-tion (15), and we also assume circular, rather than elliptical,polarization, so that E x = E y = E can be extracted as a com-mon factor. The electric field to test is now, E R (0 , t ) = ( E / { ˆ x (cos ω t − sin ω t )+ ˆ y (cos ω t + sin ω t ) } . (16)Step (vi) is achieved by setting ω t = π/
4, so thatcos ω t = sin ω t = 1 / √
2. The x component of the fielddisappears, and the field is aligned with the positive y -axis: E R (0 , t ) = ( E / √ y . (17)To see how the field rotates, we advance the time by onequarter period to t , such that ω t = ω t + π/ π/ ω t = − / √ ω t = +1 / √
2. The modified field is, E R (0 , t ) = − ( E / √ x , (18)which completes step (vii). The final step is to note that,from the observer’s viewpoint, the field has rotated counter-clockwise through one quarter turn to align with the nega-tive x -axis. The conclusion is that the presumed RCP vectorfrom equation (9) is indeed IEEE-compliant.A similar analysis (though using a different startingtime, t ) shows that the LCP vector is also IEEE-compliantas left-handed. We conclude that the helical vectors of thespherical basis used in GKK73 are IEEE-compliant and sat-isfy the IAU definition of Stokes V . The first paper of a series, Gray & Field (1995), denotedGF95 hereafter, applied the semi-classical saturation the-ory developed in Field & Richardson (1984) and Field &Gray (1988) to polarized masers, particularly for the casewhere the Zeeman splitting is large compared to the Dopplerwidth. This work is somewhat more difficult to test thanGKK73 because the electric field definition is in Cartesiancomponents. However, it is useful because the helical vectorsmay be derived from field and phase definitions, rather thanstated. We begin with equation (7), the definition of an el-liptically polarized wave in Cartesian components. It isstraightforward to transfer the entire phase factor to the y -component: lift the real part operator to yield a complexversion of the electric field, and multiply this by e − iφ x . Thereal part of this modified field is then the electric field usedin GF95 (their equation 1): E ( z, t ) = (cid:60) (cid:8)(cid:2) ˆ x E x + ˆ y E y e − iδ (cid:3) e − iY ( z,t ) (cid:9) , (19)where Y ( z, t ) is defined as before and δ = φ x − φ y , as statedin the text below equation 1 of GF95. If we assume circularpolarization to set E x = E y = E , and expand the complexexponentials, equation (19) may be developed to the form, E ( z, t ) = E(cid:60) { [ˆ x + ˆ y (cos δ − i sin δ )] (cos Y − i sin Y ) } , (20)and after multiplying out the brackets and taking the realpart, to E ( z, t ) = E(cid:60) { ˆ x cos Y + ˆ y cos( δ + Y ) } . (21)As in the GKK73 test, we proceed by setting the fixed dis-tance of z = 0 to obtain E (0 , t ) = E(cid:60) { ˆ x cos ω t + ˆ y cos( δ + ω t ) } . (22)At this point, we introduce the IEEE convention , whichrequires that for RCP radiation, the y component of thefield leads the x component: that is δ is negative and, forcircular polarization, equal to − π/
2. Inserting this value intoequation (22), we recover E (0 , t ) = E(cid:60) { ˆ x cos ω t + ˆ y sin ω t } . (23)Note that equation (23) is consistent with cosinusoidal x andsinusoidal y components in IEEE (‘source point-of-view’).As a final check, set the initial time t such that ω t = π/
2, and equation (23) reduces to the y -aligned, E (0 , t ) = E ˆ y . The time can now be advanced one quarter period, sothat ω t = ω t + π/ π , leading to E (0 , t ) = −E ˆ x . Thisis a counterclockwise rotation from the observer’s point ofview, so the electric field in GF95 is consistent with theIEEE convention. The choices made in the test above now dictate the helicalunit vector (in the spherical basis) for RCP radiation inGF95. With δ = − π/ e − iδ = i .Substitution of this result into equation (19), and setting E x = E y = E as above for circular polarization, we find thatthe RCP wave is E R ( z, t ) = E(cid:60) (cid:8) [ˆ x + i ˆ y ] e − iY (cid:9) , (24)which dictates that, for GF95,ˆ e R = (ˆ x + i ˆ y ) / √ , (25)a form identical to that used by GKK73. Rees in Kalkofen (1988) uses the optics convention; his resultfor δ was reversed to obtain the IEEE form.c (cid:13) XXXX RAS, MNRAS , 1–10
M approach to Zeeman magnetic field orientation Although both GKK73 and GF95 are IEEE-compliant intheir description of circular polarization, and use identicaldefinitions of ˆ e R , the definitions of ˆ e L are different; one canbe obtained from the other by multiplication by −
1. Fromthe point of view of circular polarization, this difference isinconsequential. We note that the GKK73 definition, ˆ e L =(ˆ x L − i ˆ y L ) √ δ = + π/
2, for LCP radiation, to obtain E (0 , t ) = E(cid:60) { ˆ x cos ω t − ˆ y sin ω t } . (26)The wave in equation (26) is y -aligned at a new start timegiven by ω t = − π/
2, and advancing it through π/ t = 0 yields an x -aligned field, demonstrating clockwiserotation from the observer’s viewpoint and therefore IEEE-left-handedness. Insertion of the corresponding phase factor, e − iπ/ = − i into equation (19), with equal real x - and y -amplitudes, then recovers the left-hand unit vector used byGKK73. Note that this form of ˆ e L is just the complex con-jugate of ˆ e R , as defined in equation (25).Note that in the paragraph above, the starting time ofthe wave was defined by ω t = − π/ ω t − π , where t is defined as in Section 4.7 ( ω t = π/ z = 0, starting at t is, from equation (19) E (0 , t − t ) = E(cid:60) (cid:8) [ˆ x − i ˆ y ] e − iω ( t − t ) (cid:9) , (27)but to start it from the same time as the RCP wave testedin Section 4.7, we eliminate t in favour of t , noting thatthis introduces a phase factor of e − iπ = −
1, transformingequation (27) to E (0 , t − t ) = E(cid:60) (cid:8) [ − ˆ x + i ˆ y ] e − iω ( t − t ) (cid:9) , (28)which yields the form of ˆ e L used in GF95, that is ˆ e L =( − ˆ x + i ˆ y ) / √ From Section 4.7.2, we note that, for a given definition ofa right-handed unit vector, two definitions of a left-handedvector are common: one is based on the same initial orienta-tion of the electric field vector (but different time origins),and the other is based on the same time (but different vec-tor orientations for the LCPs and RCPs). More importantly,the first type satisfies, ˆ e L = ˆ e ∗ R , whilst the second satisfiesˆ e L = − ˆ e ∗ R . This introduces a new layer of complexity thatdoes not affect the handedness of polarization, but shouldbe noted.We introduce here the scalar signature, s , which is thescalar product of a pair of helical vectors in the sphericalbasis: s = ˆ e R · ˆ e L . (29) s is equal to either +1 for the first type, where the vectors arecomplex conjugates, or − s = 1, but GF95 has s = − We have determined that polarization in GF95 follows theIEEE convention. However, the values of δ used in 4.7, com- bined with the definition of Stokes V in equation (7) of thatwork imply that GF95 used the non-IAU (LCP minus RCP)version of this quantity. A combination of non-IAU Stokes V and the use of the σ ± notation for transition type results ina radiative transfer equation for Stokes V (equation (24) ofGF95) that is identical in form to that in GKK73 (who useIAU Stokes V , but whose ± subscripts are reversed fromour σ ± ). The use of non-IAU Stokes V by GF95 almostcertainly led to the incorrect statement regarding polariza-tion handedness and field orientation in Section 2.2 of thatwork. Note that our equation (2) and equation (3) may beobtained from equation (24) of GF95 by making the substi-tution V GF95 = − V IAU . For GKK73 to comply with the IAU definition of Stokes V ,the right- and left-handed vectors used to represent the po-larization of the radiation must have the following associa-tion with the positive and negative helicity vectors used torepresent the response of the molecular density matrix:ˆ e + → ˆ e R ; ˆ e − → ˆ e L [GKK73] (30)In Section 4, we have considered the use of helical vec-tors to represent the handedness of radiation polarization,noting that they are often written in the form of positive andnegative helicity as above. In order to derive equations forthe transfer of the Stokes parameters, we must also describethe molecular response in terms of these same vectors. Themolecular response is used in two ways in the derivation ofthe transfer equations: once through the macroscopic polar-ization of the medium that provides source terms for thetransfer equations, and again through the Hamiltonian thatappears in the evolution equations of the molecular densitymatrix. It turns out that the Hamiltonian is adequate to fixthe helicity of the various magnetic transitions, and we willnot consider the macroscopic polarization further.By definition, the interaction Hamiltonian, comprisingits off-diagonal elements and resulting from the effect of theradiation field on the molecular electric dipole, has elementsof the form,¯ hW a,b = − E · ˆ d a,b , (31)where a and b are magnetic energy sublevels and ˆ d is thedipole operator. For the sake of example, we choose a σ + transition, noting that this has the lowest frequency in aZeeman group (see Figure 1). Note that the Zeeman groupin GKK73 is the simplest non-trivial case with m upper havingthe possible values − , , m lower = 0. Their only allowed σ + transition is thereforethe one from m upper = − m = − σ + transition is therefore,¯ hW − , = − E · ˆ d − , . (32)By comparison with equation (17) of GKK73 and our equa-tion (9), we see that the dipole for the σ + transition is right-handed, and the dipole operator may be written,ˆ d − , = ˆ d ˆ e R , (33) c (cid:13) XXXX RAS, MNRAS , 1–10
Green et al. which in the convention adopted by GKK73 correspondsto positive helicity (see equation (30) of this paper). Thisvector was originally written in a coordinate system wherethe magnetic field lies along the z -axis, and a rotation matrix(equation (19) of GKK73) was provided to rotate it ontothe system where the z -axis coincides with the direction ofradiation propagation. In the present work it is perfectlyacceptable to align the systems, as in Figure 3. Note thatequation (33) is consistent with this figure: molecules withan IEEE right-hand dipole interact with only IEEE LCPradiation in the σ + transition, since s = 1, but a right-handvector dotted onto itself is zero. V The formula derived for the molecular dipole in equation(33) is only true when the magnetic field and radiation prop-agation directions are aligned. If we reversed the magneticfield, whilst keeping the propagation direction the same, ourobserver would then see the molecular dipole for the same σ + transition represented by an IEEE left-handed helicalvector that would interact with IEEE RCP radiation. Bymeans of a rotation matrix, the general case is representedin, say, the radiation propagation frame as a linear combi-nation of both left-handed and right-handed helical vectorswith coefficients that contain functions of the angle θ be-tween the propagation and magnetic field axes. For the caseof the transfer of Stokes V , the coefficients are functions onlyof cos θ .Whatever detail is involved in obtaining it, the final ra-diative transfer equation for IAU-compliant Stokes V mustagree with the result derived in Section 5 above: for a σ + transition: IEEE LCP radiation is amplified when the mag-netic field and propagation axes are aligned (cos θ = 1). Atthis point, we note that GKK73 do not use the same σ + and σ − notation for magnetic transitions as this paper: their def-inition follows their equation (46b) on page 121, and states,‘subscripts on the Stokes parameters distinguish among thethree radiation bands by indicating the magnetic sublevelof the upper state to which each couples’. With reference toFigure 1 we see that if m lower = 0 is the only available lowerstate, as in GKK73, the upper state of the σ + transition has m upper = −
1, corresponding to a subscript − V in GKK73 that provides thenecessary information: their equation (52), line 2. Use ofthis equation requires that the magnetic field be sufficientlystrong not only to provide a good quantization axis, but alsoto split the σ − , π and σ + transitions by considerably morethan a Doppler (thermal and turbulent) width. The equationis also free of the complexities of saturation, which are notrequired to consider the sense of circular polarization. Toplace the equation in our σ − and σ + convention, we mustchange each ± symbol in the GKK73 equation with ∓ whenit refers to a radiation quantity (Stokes I and V here), butnot the one preceding the term in 2 cos θ , which is dictatedby the field and propagation geometry discussed earlier. Theresults are our equation (2) and equation (3) for σ + and σ − respectively. Note that the above operation is not the sameas a simple swap of helicities, which would swap all the ± to ∓ in the GKK73 equation and leave it invariant. LCP$$RCP$ B$ frequency$$ z"" LCP$$ RCP$ frequency$$ B$ z"" Figure 5.
Summary of observed Zeeman profiles (top) and cor-responding magnetic field directions relative to the observer (bot-tom). The IAU definition of Stokes V and the IEEE definition ofpolarization handedness are assumed. Frequency increases fromleft to right. The discussion following equation (2) and equation (3)in Section 3.2 then gives us a predominance of negativeStokes V (IEEE LCP radiation) for a σ + (GKK73 nega-tive) transition when the magnetic and propagation axesare aligned ( θ = 0). This is in accord with the discussion ofthe radiation-dipole interaction in Section 5, with Figure 3,and the classical Lorentzian derivation. We revisit the quantum mechanics and radiative transferof maser emission under the conditions of Zeeman splittingand establish the correct field orientation for an observedspectrum. Adopting the IEEE convention for right-handedand left-handed circular polarization, and the IAU conven-tion for Stokes V (right-handed circular polarization minusleft-handed circular polarization) we find (Figure 5): • A magnetic field directed away from the observer willhave right-hand circular polarization at a lower frequency(higher velocity) and left-hand circular polarization at ahigher frequency (lower velocity). • A magnetic field directed towards the observer will haveright-hand circular polarization at a higher frequency (lowervelocity) and left-hand circular polarization at a lower fre-quency (higher velocity).The results of our current analysis are consistent with theclassical Lorentzian derivations and mean that Zeeman split-ting in the Carina-Sagittarius spiral arm, as measured fromprevious studies, should be interpreted as a field directionaligned away from the observer, and thus demonstrating areal field reversal in the interstellar medium. c (cid:13) XXXX RAS, MNRAS000
Summary of observed Zeeman profiles (top) and cor-responding magnetic field directions relative to the observer (bot-tom). The IAU definition of Stokes V and the IEEE definition ofpolarization handedness are assumed. Frequency increases fromleft to right. The discussion following equation (2) and equation (3)in Section 3.2 then gives us a predominance of negativeStokes V (IEEE LCP radiation) for a σ + (GKK73 nega-tive) transition when the magnetic and propagation axesare aligned ( θ = 0). This is in accord with the discussion ofthe radiation-dipole interaction in Section 5, with Figure 3,and the classical Lorentzian derivation. We revisit the quantum mechanics and radiative transferof maser emission under the conditions of Zeeman splittingand establish the correct field orientation for an observedspectrum. Adopting the IEEE convention for right-handedand left-handed circular polarization, and the IAU conven-tion for Stokes V (right-handed circular polarization minusleft-handed circular polarization) we find (Figure 5): • A magnetic field directed away from the observer willhave right-hand circular polarization at a lower frequency(higher velocity) and left-hand circular polarization at ahigher frequency (lower velocity). • A magnetic field directed towards the observer will haveright-hand circular polarization at a higher frequency (lowervelocity) and left-hand circular polarization at a lower fre-quency (higher velocity).The results of our current analysis are consistent with theclassical Lorentzian derivations and mean that Zeeman split-ting in the Carina-Sagittarius spiral arm, as measured fromprevious studies, should be interpreted as a field directionaligned away from the observer, and thus demonstrating areal field reversal in the interstellar medium. c (cid:13) XXXX RAS, MNRAS000 , 1–10
M approach to Zeeman magnetic field orientation ACKNOWLEDGMENTS
The authors thank the anonymous referee and the manypeople who have provided thoughtful discussions. MDGacknowledges the support of the STFC under grantST/J001562/1.
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APPENDIX A: A SIMPLE TEST
The scalar or dot product of pairs of helical vectors in thespherical basis has been introduced in Section 4.7.3 to definethe scalar signature of a basis. The dot product may also beused as a very simple test to determine conformity of anybasis to the IEEE convention now that we have a vector,ˆ e R = (ˆ x + i ˆ y ) / √
2, that we know is IEEE-compliant.Unlike the real unit vectors of all common axis systems(e.g. cartesian), a helical basis vector dotted onto itself yieldsthe result zero. Compliance with the IEEE convention cantherefore be tested by taking the right-hand vector for test,ˆ e R ? , and calculating the dot product, ˆ e R · ˆ e R ? . If the resultis zero, the tested system is IEEE-compliant; if it is ±
1, thetested system is optics-compliant. All the works below weretested in this way. Table A1 presents a summary table ofthe test results. Note that it has been assumed throughoutthat the listed authors intended to follow the IAU conventionunless clearly stated otherwise, and in some cases conformity c (cid:13) XXXX RAS, MNRAS , 1–10 Green et al.
Table A1.
Electric field polarization conventions apparently usedin works discussing maser polarization and in selected generaltexts.Author(s) Convention Signature GKK73 IEEE +1GF95 IEEE − − − − See main text for definitions of shorthand notation. IEEE corresponds to ˆ e R · ˆ e R ? = 0; optics corresponds toˆ e R · ˆ e R ? = ± From evaluation of ˆ e R · ˆ e L , as discussed in Section 4.7.3. The same convention was used in the following works: Gray& Field (1994), Field & Gray (1994), Gray (2003), Gray et al.(2003). Stokes V is defined contrary to the IAU convention. to the IEEE convention on the handedness of polarizationdepends on this assumption by the present authors. A1 Deguchi & Watson (1990)
This work (DW90 for short) is concerned mostly with linearpolarization, but certainly contains sufficient information tostraightforwardly determine the convention used for circularpolarization. We note that the paper clearly states that thechoice of basis vectors is different from that used by GKK73in order to be consistent with more general work concernedwith transitions other than J = 1 −
0; specific reference ismade to Edmonds (1996).In order to be IAU compliant, equation (A20) of DW90requires that the electric field component labelled E − beRCP. However, we see from the electric field definition, equa-tion (A1), that E − is actually the coefficient of − ˆ e + , thatis there is a sign swap between the field and its associ-ated helical vector. From the above argument, and theirequation (A4a), DW90 therefore require the spherical ba-sis, ˆ e R = (ˆ x + i ˆ y ) / √
2, ˆ e L = ( − ˆ x + i ˆ y ) / √
2. This is IEEEcompliant, with scalar signature equal to −
1, and thereforein the same convention as GF95.
A2 Elitzur (1991)
This work (E91 for short) includes Elitzur (1991) and subse-quent works (Elitzur 1992, 1993). Its purpose was to extendthe work of GKK73 from J = 1 − E + corresponds to LCP, and it isassumed E + is the coefficient of ˆ e + and equivalent to ˆ e L .Under these assumptions the vectors are optics compliantand the scalar signature is +1. However, it should be notedthat Stokes V is defined contrary to IAU, with Stokes V =LCP – RCP. A3 Dinh-v-Trung (2009)
The work considered here (DvT09 for short) applied up-to-date computing power to the polarization problem, withconclusions that supported the standard model of polarized maser propagation. The definition of Stokes V is the same asin DW90 (DvT09 equation (7)), so the electric field compo-nent marked with negative helicity must be RCP. The elec-tric field definition (DvT09 equation (4)) also agrees withDW90 so we must equate ˆ e R = − ˆ e + and ˆ e L = − ˆ e − . Byinspection of DvT09 equation (1), we can see that the def-inition of ˆ e R agrees with DW90 and GF95. However, thedefinition of ˆ e L appears to be simply a negated version ofˆ e R . Presumably this is just a typographical error, and thesymbol before i should be ± rather then +. Assuming thisis so, DvT09 is IEEE-compliant. A4 Gray (2012)
In the above work (G12 for short) the Stokes parameters areset out on page 277. They have Stokes V as positive helicityfield components subtracted from negative helicity compo-nents as in DW90 and DvT09. Therefore, to conform withthe IAU convention, negative helicity components must beRCP. However, as in GKK73, but unlike DW90 and DvT09,the negative helicity (right-handed) field-component is theamplitude associated with the negative helicity unit vector,as required by equation 7.153 of G12 on page 270. We there-fore make the association, ˆ e R = ˆ e − in this case. From thedefinitions printed just above equation 7.153, and the dis-cussion above, G12 defines the right- and left-handed unitvectors as, ˆ e R = (ˆ x − i ˆ y ) / √ e L = − (ˆ x + i ˆ y ) / √ −
1. Gray (2012) is therefore internally self-consistent inhaving a field pointing away from the observer in W3(OH). c (cid:13) XXXX RAS, MNRAS000