FARADAY ROTATION MEASURE SYNTHESIS OF INTERMEDIATE REDSHIFT QUASARS AS A PROBE OF INTERVENING MATTER

The sample of quasars is selected from Bernet et al. (2008) and therefore has exactly the same Mg ii information available for every quasar from VLT UVES spectroscopy. Some sources from Bernet et al. (2008) were not observed simply because of scheduling constraints at the telescopes. With few exceptions, quasars with decl. $\gt \,10^\circ$ were observed with the Karl G. Jansky Very Large Array (VLA) while those south of this were observed with the Australia Telescope Compact Array (ATCA). Thirty-four objects were observed with VLA and 25 with ATCA. However, four objects observed with VLA and six objects observed with ATCA are not considered further for this analysis. The four VLA objects that are thrown out are 3C208, 3C281, 4C–06.35, and PKS1424–11. For those objects, it turned out that the core is too faint (and thus too low in signal-to-noise ratio) for our analysis. We believe that in prior observations the radio lobes have been confused with the core. Six ATCA objects were discarded because they are sufficiently resolved such that a phase calibration has not been possible. The decision to discard those objects was taken without any knowledge about their Mg ii absorption properties to avoid any bias in the results.

A list of the final set of 49 sources is provided in Table 1 including their position, their redshift, the redshift of their absorbers if present, and a log of observations. We declare an Mg ii absorbing system to be present if an absorption line with rest frame equivalent width ${W}_{0}\geqslant 0.1$ Å is detected along the line of sight. This deviates slightly from the earlier analysis in Bernet et al. (2008) in which a cut at W₀ = 0.3 Å has been applied. We will discuss our choice later in this paper. Consequently, the sample with interveners contains 33 objects, while the sample with clean lines of sight contains 16 objects. Similar results are found if attention is restricted only to stronger sources, albeit at reduced significance.

Table 1.  Observed Sources

Name	R.A. (J2000)	decl. (J2000)	${z}_{\mathrm{QSO}}$	Mg ii Abs.	${z}_{\mathrm{Abs}}$	Instrument	Obs. Blocka
PKS0130–17	01:32:43.4	−16:54:48	1.02	yes	0.51	ATCA	A-SB1
PKS0135–247	01:37:38.3	−24:30:54	0.84	yes	0.47	ATCA	A-SB1
PKS0139–09	01:41:25.8	−09:28:44	0.73	yes	0.50	ATCA	A-SB1
3C057	02:01:57.2	−11:32:33	0.67	no	⋯	ATCA	A-SB1
PKS0202–17	02:04:57.7	−17:01:19	1.74	yes	0.52	ATCA	A-SB1
PKS0332–403	03:34:13.7	−40:08:25	1.45	yes	1.21	ATCA	A-SB3
PKS0402–362	04:03:53.7	−36:05:02	1.42	yes	0.80	ATCA	A-SB2
PKS0422–380	04:24:42.4	−37:56:22	0.78	no	⋯	ATCA	A-SB3
PKS0426–380	04:28:40.4	−37:56:20	1.11	yes	0.56	ATCA	A-SB2
PKS0506–61	05:06:43.9	−61:09:41	1.09	yes	0.92	ATCA	A-SB3
PKS0839+18	08:42:05.1	+18:35:42	1.27	yes	0.71	VLA	V-SB1
4C+01.24	09:09:10.1	+01:21:36	1.02	yes	0.54	VLA	V-SB1
4C+02.27	09:35:18.2	+02:04:16	0.65	no	⋯	VLA	V-SB1
OK+186	09:54:56.8	+17:43:32	1.48	no	⋯	VLA	V-SB1
4C+19.34	10:24:44.8	+19:12:21	0.83	yes	0.53	VLA	V-SB1
4C+06.41	10:41:17.2	+06:10:17	1.27	yes	0.44	VLA	V-SB1
3C245	10:42:44.5	+12:03:32	1.03	yes	0.66	VLA	V-SB1
4C+20.24	10:58:17.9	+19:51:51	1.11	yes	0.86	VLA	V-SB1
PKS1111+149	11:13:58.7	+14:42:27	0.87	yes	0.65	VLA	V-SB1
PKS1127–14	11:30:07.1	−14:49:27	1.19	no	⋯	ATCA	A-SB2
PKS1143–245	11:46:08.1	−24:47:32	1.94	yes	1.25	ATCA	A-SB3
PKS1157+014	11:59:44.8	+01:12:07	1.99	yes	1.94	VLA	V-SB1
4C+13.46	12:13:32.1	+13:07:20	1.14	yes	0.77	VLA	V-SB1
4C–02.55	12:32:00.0	−02:24:05	1.05	yes	0.40	VLA	V-SB1
PKS1244–255	12:46:46.9	−25:47:48	0.63	yes	0.49	ATCA	A-SB2
ON+187	12:54:38.3	+11:41:06	0.87	no	⋯	VLA	V-SB3
4C–00.50	13:19:38.7	−00:49:41	0.89	no	⋯	VLA	V-SB3
4C+19.44	13:57:04.4	+19:19:08	0.72	yes	0.46	VLA	V-SB3
3C298	14:19:08.2	+06:28:35	1.44	no	⋯	VLA	V-SB3
PKSB1419–272	14:22:49.0	−27:27:56	0.99	yes	0.56	VLA	V-SB3
OQ+135	14:23:30.1	+11:59:51	1.61	yes	1.36	VLA	V-SB3
4C–05.62	14:56:41.5	−06:17:42	1.25	no	⋯	VLA	V-SB3
4C–05.64	15:10:53.6	−05:43:07	1.19	yes	0.38	VLA	V-SB3
4C+05.64	15:50:35.3	+05:27:11	1.42	no	⋯	VLA	V-SB3
PKS1615+029	16:17:49.9	+02:46:44	1.34	yes	0.53	VLA	V-SB3
OW-174	20:47:19.7	−16:39:06	1.93	yes	1.33	VLA	V-SB2
OX-325	21:18:10.7	−30:19:15	0.98	no	⋯	VLA	V-SB2
PKS2134+004	21:36:38.6	+00:41:55	1.94	yes	0.63	VLA	V-SB2
OX-173	21:46:23.0	−15:25:44	0.70	no	⋯	ATCA	A-SB1
4C+6.69	21:48:05.4	+06:57:39	0.99	yes	0.79	VLA	V-SB2
OX-192	21:58:06.3	−15:01:09	0.67	yes	0.39	ATCA	A-SB1
PKS2204–54	22:07:43.7	−53:46:34	1.21	yes	0.69	ATCA	A-SB1
4C–3.79	22:18:52.0	−03:35:37	0.90	no	⋯	VLA	V-SB2
PKS2223–05	22:25:47.3	−04:57:02	1.40	yes	0.85	VLA	V-SB2
PKS2227–08	22:29:40.0	−08:32:54	1.56	no	⋯	ATCA	A-SB1
4C+11.69	22:32:36.4	+11:43:50	1.04	yes	0.74	VLA	V-SB2
PKS2243–123	22:46:18.2	−12:06:52	0.63	no	⋯	ATCA	A-SB1
3C454.3	22:53:57.7	+16:08:53	0.86	no	⋯	VLA	V-SB2
PKS2326–477	23:29:17.7	−47:30:19	1.30	yes	0.43	ATCA	A-SB1

Note.

^aSee the text for specifications.

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The objects were observed in three scheduling blocks (V-SB, see Table 1) in configuration A with a maximum baseline of 36.4 km. V-SB1 was taken on 2014 March 4, V-SB2 on 2014 April 24, and V-SB3 on 2014 April 23 (Proposal ID: VLA/14A-144, PI: F. Miniati). The observations were taken using the 8 bit sampler in  the L, S, and C bands, respectively, and cover with full polarization the frequency range between approximately 1 and 6 GHz, implying a maximum frequency resolution of 2 MHz in L band and 1 MHz in the S and C bands.

CASA version 4.4.0 (McMullin et al. 2007) has been used for all calibration steps. 3C286 served as bandpass and flux calibrator in all three scheduling blocks. Since the objects are distributed over the sky, individual phase calibrators had to be chosen for each object. For nearby objects, the same calibrator has been used and bright and unresolved objects have been calibrated by themselves. The chosen phase calibrators are listed in Table 2. Sometimes sources are resolved in the L and S bands but not in the C band. In those cases, phase calibrators have been used only for the L and S bands. The phase calibrators were always observed both before and after the observation of the target. 3C286 also serves as the polarization angle calibrator (Perley & Butler 2013). The unpolarized source J1407+2827 was used to correct for polarization leakage. The residual instrumental polarization is safely below 0.3%.

Data that were obviously affected by radio frequency interference have been flagged by hand. After recalibration, further flagging has been applied by running the flagdata command in rflag mode (Greisen 2003). Altogether, around  30% of the data have been flagged for each source, mostly in the L band.

Since only the quasar itself is interesting for this work, we synthesize images just in a small window of 30$^{\prime\prime} \,\times$ 30'' around it. The images have been cleaned according to the Cotton–Schwab algorithm (Högbom 1974; Clark 1980; Schwab 1984) and applying Briggs weighting with robust parameter R = 0. To ensure that we do not get any flux leakage from other sources situated outside the synthesized window, each source has been checked to determine whether there are other bright sources within the primary beam, and this has never been found to be the case. In a few cases where the source is extended, the image sizes have been matched appropriately. L band images are made in steps of around 16 MHz and S and C Band in steps of 128 MHz. Self-calibration has been applied in all frames.

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To make the images at different frequencies comparable, all images have been smoothed to the beam size of the lowest resolution image, typically around 13. To retrieve the flux density, the brightness has been integrated over an aperture covering the FWHM of the synthesized beam centered around the maximum flux density pixel in the I frame. Subsequently, Q and U flux densities have been obtained by integrating over the same aperture in Q and U. The obtained flux densities have been converted to total flux densities assuming a two-dimensional Gaussian profile of the sources. The retrieved data (I, Q, and U parameter), as well as the derived polarization fraction Π and polarization angle χ, are shown in the Columns (1)–(4) of Figures 1–3.

The ATCA observations took place on 2014 May 3 (A-SB1, see Table 1) and 2013 May 14 (A-SB2) and 15 (A-SB3) in configuration 6C corresponding to a maximum baseline of 6 km. The observed frequency range was 1.1–3.1 GHz (16 cm band), 4.0–6.0 GHz (4 cm band) and 8.8–10 GHz (4 cm band) with a resolution of 1 MHz (Proposal ID: C2769, PI: M. L. Bernet). Each source has been observed with at least three cuts in the uv-plane.

The reduction package MIRIAD (Sault et al. 1995) has been used to carry out the calibration. For sources in A-SB1, PKSB1934–638 served as the bandpass calibrator in the 16 cm band and PKSB1921–293 in the 4 cm band, while, for sources in A-SB2 and A-SB3, PKS0823–500 served as the bandpass calibrator. Furthermore, PKSB1934–638 is used as the flux calibrator for all sources. We observed PKSB2326–477 and PKSB2005–489 repeatedly during A-SB1 and PKS1903–802 during A-SB2 and A-SB3 for the phase and polarization leakage calibration. However, since the phase calibration solutions turned out to be unsatisfying, we relied solely on self calibration for all sources. This is the reason why some sources were discarded. The pgflag command has been used to auto-flag corrupted data. Some manual flagging has been applied afterwards.

I, Q, and U Images of 2' × 2' frames around the objects are synthesized in steps of 16 MHz over the whole observed frequency band applying natural weighting. Cleaning has been carried out with the clean command in any mode. All frames have been smoothed to the same resolution, typically around $10^{\prime\prime}$, and it has been ensured that they contain only one single unresolved source. The flux density of that source has been determined by measuring the maximum brightness of the frames. The data points of the 4 cm band images have been binned in steps of 128 MHz to have similar data spacings as the VLA objects.

The retrieved data (I, Q, and U) as well as the derived polarization fraction Π and polarization angle χ are shown in the Columns (1)–(4) of Figures 1–3. We conclude that the structure of I, Q, and U is comparable between sources observed at VLA with sources observed at ATCA, ensuring the consistency of our sample.

RM Synthesis was carried out using the code provided by Brentjens & de Bruyn (2005). The RM Synthesis is calculated with uniform weighting and relative to ${\lambda }^{2}=0$. Subsequently, the RM-CLEAN algorithm has been applied, following Heald et al. (2009). The effects of RM-CLEAN are shown in Figure 4 for four objects. PKS0839+18 and PKS2326–477 are sources with simple FD distributions that were observed with the VLA and ATCA, respectively. 4C+01.24 and PKS0135–247 represent sources with more complex FD distributions from the VLA and ATCA, respectively. The second row shows the dirty $F(\phi )$, that is before RM-CLEAN is applied, and the third row shows the clean $F(\phi )$ after the application of RM-CLEAN.

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The first row shows the RM Spread Functions (RMSF) of the objects. The RMSF is the function with which the true FD distribution is convolved due to the finite sampling of the data in ${\lambda }^{2}$ space. After running RM-CLEAN, the RMSF corresponds to a Gaussian, whereby its FWHM can be estimated as (Brentjens & de Bruyn 2005)

$$\begin{eqnarray}&&\delta \phi \approx \displaystyle \frac{2\sqrt{3}}{{\rm{\Delta }}{\lambda }^{2}},\end{eqnarray} \tag{ 6 }$$

where ${\rm{\Delta }}{\lambda }^{2}$ is the observed frequency bandwidth in ${\lambda }^{2}$ space. For ease of comparison, later on we introduce

$$\begin{eqnarray}&&{\sigma }_{\mathrm{RMSF}}=\displaystyle \frac{\delta \phi }{2\sqrt{2\mathrm{ln}2}},\end{eqnarray} \tag{ 7 }$$

defined as the standard deviation of the clean Gaussian RMSF. ${\sigma }_{\mathrm{RMSF}}$ corresponds to the "resolution" of the data in ϕ space. The ${\sigma }_{\mathrm{RMSF}}$ for the VLA data is around 17 rad m⁻² and, for the ATCA data, it is around 24 rad m⁻².

Despite some known shortcomings of the RM-CLEAN method to accurately reconstruct complex FD distributions (Sun et al. 2015), we dispense with more sophisticated but complicated methods (e.g., Frick et al. 2010; Farnsworth et al. 2011; Li et al. 2011; O'Sullivan et al. 2012; Schnitzeler & Lee 2015). We regard the method adopted to be sufficient for our purposes since our analysis aims primarily to separate sources with simple FD distributions from those with complex ones. As will become clear later in the paper, our conclusions do not depend on small components in the FD distribution.

Historically, it has been conventional to use the Stokes parameter Q and U as input for the RM Synthesis, according to the definition in Equation (4). Recently, however, the fractional parameter q = Q/I and u = U/I has been utilized (Anderson et al. 2015). There are good reasons to do that, especially when one is dealing with steep spectral index sources. However, since our sources mostly have flat spectra, we do not expect the differences of the two methods to be severe. Indeed we have tried out both methods and it turns out that the conclusions we draw in this paper hold whatever method we use. For the presentation of the results in this paper, we choose to stick with the conventional method, i.e., utilizing Q and U, and mention the results with q and u.

The obtained FD distributions $F(\phi )$ are shown in the panels of Column (5) of Figures 1–3 for all 49 sources. Although the FD distributions are synthesized between −1500 and 1500 rad m⁻², we show only the range between −750 and 750 rad m⁻² since we do not detect any significant signal outside of this window. We adopt the Galactic RM contribution estimates from Oppermann et al. (2015) and shift the overall FD distributions accordingly.

The polar plots in the panels of Column (6) represent the complex FD distribution ${\boldsymbol{F}}(\phi )$. In this, the azimuthal angle represents the initial phase ψ, i.e., the polarization angle of a particular component at infinite frequency (see Equation (5) above) while the radius is a measure of the amplitude that was plotted in Column (5). We obtain continuous curves since both $F(\phi )$ and $\psi (\phi )$ are continuous. The initial phase ψ is defined between −$\pi /2$ and $\pi /2$ and thus, for plotting purposes, ψ is multiplied by two.

The FD distribution $F(\phi )$ decomposes the observed flux density in ϕ, i.e., it describes the amount of linearly polarized flux density that has undergone a certain amount of Faraday Rotation due to lying at a certain Faraday Depth. Thus, for a homogeneous screen lying in front of a simple source, the FD distribution would ideally be a delta function. An inhomogeneous screen would result in a broader or more complicated FD distribution because different parts of the background source would have passed through different Faraday depths ϕ. In this work, we are interested in extracting information on the inhomogeneity of foreground screens and therefore the FD distribution is what we are ultimately interested in.

However, from the FD distribution alone, we do not know if structure within $F(\phi )$ is caused by variations within an intervening system, somewhere along the line of sight, or by having a source that is itself Faraday thick. Sources are called Faraday thick if they have a range of ϕ due to magnetized plasma intrinsic to the sources. This can occur in two ways, either through internal Faraday dispersion or through differential Faraday Rotation (Sokoloff et al. 1998). Internal Faraday dispersion is caused by intrinsic inhomogeneous screens. Effectively, the source is composed of different sub-components each with their own ϕ. Differential Faraday Rotation is caused if a source is extended along the line of sight such that flux that is emitted from the far side of the object undergoes more rotation than flux that is emitted from the near side.

To discriminate between intrinsic inhomogeneity effects and intervener inhomogeneity effects, $\psi (\phi )$ can be very useful. As mentioned already in Section 1 the initial phase represents the average polarization angle of the flux density at a certain ϕ. Thus, if all the emission has the same origin, then $\psi (\phi )$ should be constant, independent of how complex the FD distribution might be from the intervener system. If, however, $\psi (\phi )$ is not constant, then it can be concluded that the emission at the different ϕ must have different spatial origins within the source. Correspondingly, it becomes very likely (but not absolutely proven) that the variation of ϕ is also intrinsic to the source. The case of constant $\psi (\phi )$ corresponds to a radial locus of ${\boldsymbol{F}}$ in a polar representation, i.e., a linear feature extending out from the central origin. There are numerous cases of this in Figures 1–3 (e.g., PKS0839+18, 4C+13.46 and 4C+05.64).

To illustrate a more complex example, consider again the case of a source (with intrinsic magnetic fields) that is extended along the line of sight. The emission coming from the far side of the source can be produced with a different polarization angle than emission that is coming from the near side and therefore can have different initial phases. Eventually, one would expect an ellipse-like structure in the polar plots of the complex FD distribution in which the initial phase varies smoothly with the Faraday Depth. Analogously, one would also get an ellipse-like structures if there are unresolved spatial polarization structures in the extended source as well as an spatially inhomogeneous FD screen in front of it. Examples of ellipses in Figures 1–3 are 4C+02.27, OW-174, and 4C–05.62.

More complex situations are also possible. A source consisting of a number of discrete components within the telescope beam, each with a very narrow range of ϕ, will exhibit the corresponding number of peaks in $F(\phi )$ (in Column (5)) and radial spikes in ${\boldsymbol{F}}(\phi )$ (in Column (6)) producing a complex amoeba-like structure in the latter. Examples are given by 4C–00.50, PKS1143–247, and PKS0506–61. On the other hand, a simple (but extended) background source that undergoes Faraday Rotation by a number of discrete ϕ within an inhomogeneous foreground screen would consist of multiple peaks in $F(\phi )$ (in Column (5)) but the radial spikes in ${\boldsymbol{F}}(\phi )$ (in Column (6)) would be aligned. Interestingly, we see no such sources in our sample, a point to which we will return later.

Overall, we see that there is a wide variation in complexity of $F(\phi )$ and ${\boldsymbol{F}}(\phi )$ within our sample as shown in Figures 1–3. However, despite this clear variety, we also see that each object usually has a pronounced dominant Gaussian-like component in which a large fraction of the total flux density is contained. This is in agreement with the observations of low redshift quasars in Anderson et al. (2015). We will refer to this component as the "primary component" throughout this paper. All other significant components will be referred to as "secondary components." Roughly two-thirds of our sample possess significant secondary components. Generally speaking, the structure of the FD distribution is broadly comparable between those sources observed at VLA and those observed at ATCA, except for the different widths of the RMSF.

As is clear in Figures 1–3, essentially every object (with the possible exception only of PKS2134+004, see below) has an FD distribution that is dominated by a single pronounced primary component at a well-defined ${\phi }_{\max }$. We define ${\tilde{\phi }}_{\max }$ as the peak position as observed, and ${\phi }_{\max }$ as the peak position after shifting the FD distribution by a uniform ${\rm{\Delta }}\phi$ corresponding to an estimate of the Galactic foreground according to Oppermann et al. (2015).

It is therefore of interest to compare the ${\phi }_{\max }$ for each source with the RM_Kron of Kronberg et al. (2008) used in Bernet et al. (2008). These were based on polarization angle measurements at just a few distinct frequencies. As discussed in Bernet et al. (2012), there are some caveats to this traditional way of measuring RM, especially if the sources are significantly depolarized. These problems are largely circumvented in RM Synthesis.

In Figure 5, we compare the primary component peak ${\tilde{\phi }}_{\max }$ of the FD distribution, before any Galactic RM correction, to the RM_Kron, which were used (similarly uncorrected) in Bernet et al. (2008). For about three quarters of the objects, the agreement between these quite independent measurements is very good with a random dispersion of roughly 8 rad m⁻². For 12 objects, there is a significant discrepancy. Many of the objects with strong $| {\mathrm{RM}}_{\mathrm{Kron}}| \gt 50\,{\rm{rad}}$ m⁻² are found to have rather smaller $| {\tilde{\phi }}_{\max }|$. There are three objects with very large discrepancies. The case of PKS2134+004 is interesting in that the dominant peak is clearly at very high $\tilde{\phi }$. This source shows extraordinarily complex FD structures (Figure 3, Row 6). The position angle $\chi ({\lambda }^{2})$ plot for this source emphasizes the difficulty of measuring an RM, for such a source, from sparsely sampled data. In the main, however, this comparison is reassuring that the original RM measurements used in Bernet et al. (2008) were meaningful.

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Many years ago, Burn (1966) calculated the relation between depolarization and a dispersion in the Faraday Depth to be of the form

$$\begin{eqnarray}&&{\rm{\Pi }}({\lambda }^{2})={{\rm{\Pi }}}_{0}\exp (-2{({\sigma }_{\mathrm{Burn}}{\lambda }^{2})}^{2})\end{eqnarray} \tag{ 8 }$$

where ${\rm{\Pi }}({\lambda }^{2})$ is the polarization fraction as a function of ${\lambda }^{2}$, ${{\rm{\Pi }}}_{0}$ is the intrinsic polarization fraction before the Faraday depolarization, which is assumed to be constant with λ, and ${\sigma }_{\mathrm{Burn}}$ is the standard deviation of a Gaussian FD distribution. Subsequently, various modifications to this simple model have been proposed to better fit observations (Tribble 1991; Rossetti et al. 2008; Bernet et al. 2012).

All these models produce a monotonic decrease of ${\rm{\Pi }}({\lambda }^{2})$ with λ. The measured polarization of many of our sources, however, clearly shows a more complex structure in ${\rm{\Pi }}({\lambda }^{2})$. We will explore possible reasons for this in Section 5.4. An extensive overview of depolarization due to Faraday screens can be found in the appendix of Farnes et al. (2014).

We want to test the suggestion in Bernet et al. (2012) that intervening Mg ii absorption along the line of sight also produces detectable depolarization effects. We therefore need to quantify the depolarization in the radio spectra. However, the variety and complexity of the polarization spectra (Column (3) in Figures 1–3) makes it impossible to represent the depolarization with a single parameter. We therefore defined several parameters. Although some of these may appear somewhat ad hoc, this was done blind to the presence of interveners.

We first define DP_Quart and DP_Half as

$$\begin{eqnarray}&&{\mathrm{DP}}_{\mathrm{Quart},\mathrm{Half}}=1-\displaystyle \frac{{{\rm{\Pi }}}_{2}}{{{\rm{\Pi }}}_{1}},\end{eqnarray} \tag{ 9 }$$

where ${{\rm{\Pi }}}_{1}$ is the median Π in the first quarter of the observed ${\lambda }^{2}$ band, i.e., in the short wavelength end, and ${{\rm{\Pi }}}_{2}$ the median Π in the last quarter for DP_Quart, i.e., in the long wavelength end. For DP_Half, ${{\rm{\Pi }}}_{1}$ is the median Π in the first half of the observed ${\lambda }^{2}$ band and ${{\rm{\Pi }}}_{2}$ the median Π in the other half. By definition DP_Quart and DP_Half yield negative values for re-polarizing sources.

We also define DP${}_{\mathrm{Min}/\mathrm{Max}}$ as

$$\begin{eqnarray}&&{\mathrm{DP}}_{\mathrm{Min}/\mathrm{Max}}=1-\displaystyle \frac{{{\rm{\Pi }}}_{\min }}{{{\rm{\Pi }}}_{\max }},\end{eqnarray} \tag{ 10 }$$

where ${{\rm{\Pi }}}_{\min }$ and ${{\rm{\Pi }}}_{\max }$ are the minimum and maximum observed Π, respectively, wherever they occur in the observed λ range. We also define $\mathrm{DP}{^{\prime} }_{\mathrm{Min}/\mathrm{Max}}$, which is similar to DP${}_{\mathrm{Min}/\mathrm{Max}}$ except that ${{\rm{\Pi }}}_{\min }$ is the minimum polarization at a longer wavelength than the maximum ${{\rm{\Pi }}}_{\max }$. DP_Quart, DP_Half, DP${}_{\mathrm{Min}/\mathrm{Max}}$, and $\mathrm{DP}{^{\prime} }_{\mathrm{Min}/\mathrm{Max}}$ are completely phenomenological parameters.

Lastly, we introduce ${\sigma }_{\mathrm{Burn}}$ and $\sigma {^{\prime} }_{\mathrm{Burn}}$ as two further depolarization parameters. The ${\sigma }_{\mathrm{Burn}}$ parameter is defined as in Equation (8) and is obtained by fitting this relation to ${\rm{\Pi }}({\lambda }^{2})$ over the whole observed wavelength range while the $\sigma {^{\prime} }_{\mathrm{Burn}}$ parameter is obtained by only fitting ${\rm{\Pi }}({\lambda }^{2})$ at wavelengths longer than ${{\rm{\Pi }}}_{\max }$. The latter is motivated by the conjecture that depolarization might be mainly caused by a simple dispersion in Faraday Depth while other more complex behavior of the polarization structure might be caused by other processes. The fits are carried out by using the method of least squares. Although ${\sigma }_{\mathrm{Burn}}$ and $\sigma {^{\prime} }_{\mathrm{Burn}}$ might seem like the most appropriate parameters if the FD distribution was Gaussian, they might be less appropriate for the more complex polarization curves. The values of the six depolarization parameters for each object are listed in Table 3.

As with depolarization, it is not trivial to identify the most suitable parameters to characterize the structure in the FD distributions due to the variety of the observed distributions (see Figures 1–3, Column (5)). As in the previous section, we approached this problem by defining several parameters. To preserve objectivity, we again defined all the parameters in this section blind to the presence of interveners in the sources.

An important point is that, because RM Synthesis behaves like a Fourier Transform, random "white" noise in the input Q and U spectra will produce a non-zero contribution to the FD distribution $F(\phi )$ that should be independent of ϕ and extends out to the ϕ corresponding to the spectral resolution of the input data, in our case ±1500 rad m⁻². In practice, this noise contribution to the FD distribution will itself have structure due to noise in the data. To avoid including this spurious contribution into our characterization of the structure in the FD distribution, the following thresholding scheme was implemented. We first apply an iterative sigma-clipping algorithm (clipping at 3σ) to $F(\phi )$ over the whole ±1500 rad m⁻² range to estimate the background mean and standard deviation of $F(\phi )$ that arises from this noise. We then identify real signals in $F(\phi )$ as being those regions that exceed this mean plus 5σ, in the sense that we can be reasonably confident that $F(\phi )$ in these regions is not coming from noise. These are the parts of ϕ space that we want to use to characterize the FD distribution. This threshold is indicated by the magenta lines in Column (5) of Figures 1–3. For each region above the threshold, we then extend in each direction out to the ϕ at which $F(\phi )$ first crosses the mean background level. All other regions are set to zero, producing a thresholded $\hat{F}(\phi )$ function that can then be parameterized as described below.

The first parameter is simply the second moment of the modified FD distribution $\hat{F}(\phi )$

$$\begin{eqnarray}&&{\sigma }_{\mathrm{FD}}^{2}=\displaystyle \frac{\int \hat{F}(\phi ){(\phi -{\mu }_{\mathrm{FD}})}^{2}d\phi }{\int \hat{F}(\phi )d\phi },\end{eqnarray} \tag{ 11 }$$

where ${\mu }_{\mathrm{FD}}$ is the usual first-moment of the FD distribution. Because RM Synthesis yields a discrete FD distribution at equidistant ${\phi }_{i}$, this integral is conveniently done as a sum over the non-zero data points. This parameter has also been used in Anderson et al. (2015).

The second parameter is the reversed Gini coefficient G. The reversed Gini coefficient is simply defined (for our equidistant data points) as

$$\begin{eqnarray}&&G=1-\displaystyle \frac{1}{2{\mu }_{\mathrm{FD}}n(n-1)}\displaystyle \sum _{i}^{n}\displaystyle \sum _{j}^{n}| \hat{F}({\phi }_{i})-\hat{F}({\phi }_{j})| ,\end{eqnarray} \tag{ 12 }$$

where the sum is carried out over the full range of ϕ so that n is the same for all objects. This means that objects can be directly compared to each other, even though the absolute value of G depends on the range of ϕ (i.e., on n). The Gini coefficient is generally used to measure the concentration of a distribution, e.g., in economics to measure the concentration of wealth within a population (Gini 1912) or in astronomy to measure the luminosity concentration in a galaxy (Abraham & van Bergh 2003). G is bounded between 0 and 1 whereby 1 means equal distribution and 0 means total concentration. We choose the reversed Gini coefficient because we want to maintain the convention that non-concentrated FD distributions yield large parameters. In practice, since many sources are dominated by a single component, G correlates quite well with ${\sigma }_{\mathrm{FD}}$.

We also introduce a "coverage parameter" C. This parameter is motivated by the suggestion that interveners may only partially cover the source (Rossetti et al. 2008; Mantovani et al. 2009; Bernet et al. 2012). In this scenario, we could interpret that the primary component in the FD distribution represents the flux that is not covered by the intervening screen. We model the primary component by fitting a Gaussian to it, using the method of least squares, and subtracting that fit from $\hat{F}(\phi )$ distribution to yield ${\hat{F}}_{\mathrm{cov}}(\phi )$. Our coverage parameter is then determined as

$$\begin{eqnarray}&&C=\displaystyle \frac{\int {\hat{F}}_{\mathrm{cov}}(\phi )d\phi }{\int \hat{F}(\phi )d\phi }.\end{eqnarray} \tag{ 13 }$$

We obtain values of C up to around 40% in our sample.

With the parameters above, we have attempted to describe the variety of complexity that we clearly can see. As a final subjective classification, we ranked the objects in order of perceived complexity. Two of us (KSK and SJL) independently ranked the FD distributions. Despite the obvious subjectivity, these rankings agreed well, except for the roughly one-third of the sources that have strikingly simple and therefore very similar FD distributions. However, the Kolmogorov Smirnov (KS) test, which we will be using throughout this paper, is relatively robust against permutations within only the low ranked fraction of the objects. This method is to be regarded mostly as a test of whether the quantitative parameters we used do indeed represent the perceived complexity. This subjective classification was again done blind to the presence of Mg ii absorption.

For reasons that will become clearer later (see Section 5.4), we introduce one more parameter ${\sigma }_{\mathrm{PC}}$, which has been defined a posteriori. The ${\sigma }_{\mathrm{PC}}$ parameter is obtained by fitting a Gaussian to the primary component. The range for this fit is where the FD distribution $F(\phi )$ stops declining on either side of the primary component.

To correct for the intrinsic spread function (RMSF) in ϕ that arises from the finite wavelength bandwidth of the radio data, which is different for the VLA and ATCA observations, we subtract in quadrature the predicted RMSF from the fitted ${\sigma }_{\mathrm{fit}}$ as follows:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{PC}}=\sqrt{{\sigma }_{\mathrm{fit}}^{2}-{\sigma }_{\mathrm{RMSF}}^{2}},\end{eqnarray} \tag{ 14 }$$

where ${\sigma }_{\mathrm{RMSF}}$ is defined in Equation (7). The obtained values of all five FD distribution parameters for each object are listed in Table 3.

In Bernet et al. (2008), it was claimed that there was an association between higher $| \mathrm{RM}|$ values and the presence of strong (rest equivalent width above 0.3 Å) Mg ii absorption in the optical spectrum. This was based on a KS test on the distribution of $| \mathrm{RM}|$ for sources with and without such absorption. Here we reproduce this analysis but now using our new $| {\phi }_{\max }|$ instead of $| {\mathrm{RM}}_{\mathrm{Kron}}|$. $| {\phi }_{\max }|$ is the modulus of the Faraday Depth at which the FD distribution peaks, shifted by the Oppermann et al. (2015) estimates of the Galactic contribution to ϕ. We use the one-tailed Kolmogorov–Smirnov to test the hypothesis that objects with Mg ii absorption along the lines of sight have enhanced $| {\phi }_{\max }|$ (i.e., our null hypothesis is that there is no such association or that clean lines of sight have enhanced $| {\phi }_{\max }|$). Recall that objects with ${\phi }_{\max }\gt 80\,{\rm{rad}}$ m⁻², i.e., PKS0506–61, PKS2134+004, and 4C+6.69, are excluded from the analysis since we believe they are strongly affected by Galactic effects (see Section 4).

Unlike Bernet et al. (2008), we adopt a weaker absorption cut at W₀ = 0.1 Å and get $p=17 \%$, represented in Figure 8. We will use this cut throughout the remainder of the paper because this cut turns out to be more significant later on when we analyze the distribution structure of Faraday Depth, indicating that weak absorbers also affect the FD distribution. However, also using the same equivalent width cut at 0.3 Å in Mg ii absorption, we do not see a signal and obtain a p-value of 23%. Using fractional q and u for RM Synthesis as discussed in Section 2.4 makes effectively no difference to the significance level of this test for $| {\phi }_{\max }|$.

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With our new data, the correlation of Mg ii with $| {\phi }_{\max }|$ is less significant than found with RM in Bernet et al. (2008). They found a significance level of 92.2% in a two-tailed KS test, without even correcting for Galactic RM.

This difference may be partly due to the fact that we are only analyzing a subset of their sample. On the other hand, it could also reflect differences between the use of ${\phi }_{\max }$ derived from RM Synthesis (as here) and the use of single RM values that are derived from individual sparsely sampled polarization data (as in Bernet et al. 2008). Indeed, when we do the KS test with RM_Kron for our sample (excluding PKS0506–61, PKS2134+004, and 4C+6.69 as with our ${\phi }_{\max }$ analysis), we obtain a somewhat stronger result with $p=13 \%$. This stronger significance is driven mostly by the objects with $| {\mathrm{RM}}_{\mathrm{Kron}}| \gt 50\,{\rm{rad}}$ m⁻², which is where we see discrepancies between ${\phi }_{\max }$ and RM_Kron, as mentioned in Section 3.1. This difference between ${\phi }_{\max }$ and RM_Kron will be explored further in a future paper.

Prompted by the statistical editor, we also have performed the Anderson–Darling (AD) 2-sample test for all our KS test results (here and later). It turns out that the AD test results are always slightly more significant but altogether consistent with the KS test results. Therefore, we remain with the KS test throughout this paper.

It should be noted that the sources with intervening absorption systems are generally at slightly higher redshifts (a difference of around 0.2 in their respective medians) and so the simple dependence of ϕ on wavelength would produce, all other things being equal, a lower observed dispersion for the more distant systems, i.e., opposite to what is seen in Figure 8.

We now look for correlations between the depolarization parameters and the presence of Mg ii absorption lines along the line of sight. As before, we apply the one-tailed KS test with the null hypothesis that sources with Mg ii absorption along the lines of sight are drawn from the same distribution as objects with clean lines of sight or that objects with clean lines of sight are more strongly depolarized.

The strongest signal is achieved with DP${}_{\mathrm{Min}/\mathrm{Max}}$ for which the KS test yields a p-value of only $p=0.1 \%$. Figure 9 shows the distribution functions of DP${}_{\mathrm{Min}/\mathrm{Max}}$. As already mentioned in Section 4, if objects at high Galactic latitude $| b| \gt 55^\circ$ are excluded, the signal increases to a nominal $p=0.01 \%$.

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Also, DP${^{\prime} }_{\mathrm{Min}/\mathrm{Max}}$ yields a strong result with $p=1.3 \%$. ${\sigma }_{\mathrm{Burn}}^{\prime }$ is not as robust, due to the complex behavior of ${\rm{\Pi }}({\lambda }^{2})$ in many sources, and for 10 objects the fit either fails to converge or yields negative ${\sigma }_{\mathrm{Burn}}$, which is unphysical. Once these objects are excluded from the KS test, however, $\sigma {^{\prime} }_{\mathrm{Burn}}$ yields $p=0.5 \%$.

To evaluate the real significance of the obtained p-values (here and elsewhere in the paper) we have randomly re-scrambled objects with and without Mg ii absorption. The re-scrambling has been realized by randomly drawing, without replacement, 16 objects and declaring them artificially to be objects with clean lines of sight while the remaining 33 objects are declared as objects with interveners. Subsequently, the KS test has been carried out to obtain p-values. The distribution of p-values (after 10,000 such draws) showed that the p-value estimates are conservative in the sense that p-values smaller than $p=0.1 \%$ happen in 0.09%, $p=1.3 \%$ in 0.6%, and $p=0.5 \%$ in 0.2% of the realizations. This check shows that the p-values of the KS tests are conservative.

Somewhat weaker but still significant results are also achieved for DP_Half and ${\sigma }_{\mathrm{Burn}}$ with $p=3.9 \%$ and $p=9.0 \%$. As with $\sigma {^{\prime} }_{\mathrm{Burn}}$, the fit of ${\sigma }_{\mathrm{Burn}}$ fails for nine objects and those are excluded from the ks test for ${\sigma }_{\mathrm{Burn}}$. For DP_Quart, we see little or no effect with $p=15 \%$.

It is clear that we obtain rather different p-values with the six depolarization parameters and we will discuss this further in Section 5.4. Nonetheless, altogether, our results demonstrate that there is a clear and highly significant correlation between the presence of intervening Mg ii absorption systems and depolarization.

Farnes et al. (2015) compared fractional polarization spectral indices to the presence of Mg ii absorption with 41 objects and could not see any correlation between polarization structure and presence of interveners. However, recall that our sources are initially selected to be compact while the sample in Farnes et al. (2015) reduces to 15 when only flat spectrum sources are selected. Furthermore, in Farnes et al. (2015), only a handful of data points along the spectrum has been available, which could lead to an imperfect description of the polarization structure given the complexity we can find for some sources.

The previous section demonstrated that depolarization in the radio spectra is clearly statistically associated with the presence of intervening Mg ii absorption along the line of sight. Since, generically, depolarization reflects the presence of different ϕ coming from different parts of the source, we would then expect to see correlations between the presence of interveners and various parameters that capture the range of ϕ in a given source.

Armed with the quantitative parameters defined in Section 3.3, we then applied the KS tests. As before, we adopt the null hypothesis that the parameters are equally distributed between objects with and without intervening absorption systems or are enhanced for objects without interveners.

We find no significant cause to reject the null hypothesis, obtaining $p=17 \%$ for ${\sigma }_{\mathrm{FD}}$, $p=12 \%$ for G, $p=33 \%$ for C, and $p=24 \%$ for the subjective ranking. Figure 10 shows the distributions of the four parameters and it is fairly clear that there is no significant correlation between them and the presence of intervening absorbers. The results remain insignificant when the fractional q and u are used for RM Synthesis with $p=15 \%$ for ${\sigma }_{\mathrm{FD}}$, $p=32 \%$ for G, and 55% for C.

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These null results were surprising. We clearly see the connection between interveners and depolarization in Section 5.2, but cannot then associate the complexity in the FD distribution, which we would have expected was the cause of the depolarization, with the presence of interveners. Furthermore, we actually see very little correlation between our parameterizations of FD structure and depolarization.

This result then led us to re-examine the links between the FD distribution and depolarization, to isolate that feature of the FD distribution that is most strongly causing depolarization, and then to construct a further FD parameter that is, finally, clearly associated with the presence of intervening systems in our sample. This is the subject of the next two Sections 5.4 and 5.5. Furthermore, we are able to argue that the overall structure of the FD distribution examined in this section is actually likely reflecting other properties of the sources. This is briefly examined in Section 5.6.

As commented at the end of the previous section, we found a surprising disconnect between the evident richness of structure in the FD distribution $F(\phi )$ and the presence of depolarization signatures in the overall polarization spectrum. The richness of the polarization structure was also surprising. As discussed already, we might expect to see simple monotonically decreasing polarization with increasing wavelength (for a simple Gaussian FD distribution), but instead we see a very wide range of behavior (as in Column (3) of Figures 1–3).

The polarized flux density of a given source at a given wavelength represents the vector sum of the complex representations of the different ϕ components, each of which rotates at a speed (in ${\lambda }^{2}$ space) that is proportional to ϕ. It is worth distinguishing between the effects on $P({\lambda }^{2})$ of a few components in $F(\phi )$ that are widely separated in ϕ, referred to as "gross structure" in the following, and the effects of very closely spaced, or continuous, structure in $F(\phi )$, as, for example, in the Gaussian width of a particular component, referred to as "fine structure." The former causes large variations in the polarized flux density with wavelength as the small number of polars rotate around each other producing an oscillatory behavior in the total amplitude, i.e., in the polarized flux density. In contrast, the fine structure in ϕ within a feature produces a more steady decrease in polarization as the continuous distribution of Faraday Rotation causes a progressive cancellation of polarized flux.

The FD distributions of many sources in Figures 1–3 show multiple discrete components, i.e., gross structure, and the parameters which we constructed in the previous section, including our own subjective ranking, were based primarily on the presence of these multiple discrete components rather than the width of individual components.

Further evidence for the distinction between multiple components and their widths (i.e., between gross and fine structure) comes from the comparison of the ${\sigma }_{\mathrm{FD}}$ values to the parameters obtained from the depolarization curves (i.e., ${\sigma }_{\mathrm{Burn}}$ or $\sigma {^{\prime} }_{\mathrm{Burn}}$). Although we would have expected those parameters to trace the same physical quantity, i.e., the dispersion of the FD distribution, we obtain ${\sigma }_{\mathrm{FD}}$ to be an order of magnitude larger than the depolarization parameters. This could be explained by arguing that ${\sigma }_{\mathrm{FD}}$ describes gross structures while ${\sigma }_{\mathrm{Burn}}$ and $\sigma {^{\prime} }_{\mathrm{Burn}}$ describe fine structures. The ${\sigma }_{\mathrm{PC}}$ parameter is sensitive to these fine structure effects because it measures the widths of the primary components. We will show later that they are of similar sizes as ${\sigma }_{\mathrm{Burn}}$ and $\sigma {^{\prime} }_{\mathrm{Burn}}$.

We explore the effects of both discrete components and the Gaussian width on the polarization $P(\lambda )$ using a simple toy model. The toy model consists of N = 1000 FD cells, which are each associated with ${\phi }_{i}$ according to an underlying FD distribution, assuming that all cells have the same initial phase ψ and also assuming unit total flux density, I, so that $P(\lambda )={\rm{\Pi }}(\lambda )$. The polarization $P({\lambda }^{2})$ is then determined by

$$\begin{eqnarray}&&P({\lambda }^{2})=\sqrt{U{({\lambda }^{2})}^{2}+Q{({\lambda }^{2})}^{2}},\end{eqnarray} \tag{ 15 }$$

where

$$\begin{eqnarray}&&U({\lambda }^{2})=\displaystyle \frac{1}{N}\displaystyle \sum _{i}^{N}\sin (2{\phi }_{i}{\lambda }^{2}),\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&Q({\lambda }^{2})=\displaystyle \frac{1}{N}\displaystyle \sum _{i}^{N}\cos (2{\phi }_{i}{\lambda }^{2}).\end{eqnarray} \tag{ 17 }$$

Figure 11 shows the resulting depolarization structure of a few simple models. The left panel corresponds to a model with one ϕ component, the middle panel to a model with two components where the secondary component carries 10% of the total flux density. And the right panel to a model with three components where the first secondary component carries 20%, and a second 10%, of the total flux density. All ϕ components have been convolved with a Gaussian of different widths in ϕ, namely ${\sigma }_{\mathrm{disp}}=0$ (blue), ${\sigma }_{\mathrm{disp}}=5\,{\rm{rad}}$ m⁻² (green), ${\sigma }_{\mathrm{disp}}=10\,{\rm{rad}}$ m⁻² (red), and ${\sigma }_{\mathrm{disp}}=20\,{\rm{rad}}$ m⁻² (cyan), respectively.

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We see that highly non-monotonic and complex polarization structure is produced by adding just a few additional components. However, the overall depolarization is mainly driven by the convolution with the Gaussian rather than by these multiple components. Multiple discrete components in contrast tend to add oscillatory features to the polarization structure. The small amplitude of the oscillations reflects the small fractions of the flux density in the secondary components. In contrast, the convolution leads to phase dispersions affecting all of the polarized flux density.

We can now return to the different parameterizations of the depolarization introduced in Section 3.2 and examine which of them best reflects the fine structure ${\sigma }_{\mathrm{disp}}$, recalling that these were correlated to different degrees with the presence of Mg ii intervening absorption. We use the same type of simple models, but now construct 6000 models by varying the positions of the secondary component(s) relative to the primary component (between zero and 200 rad m⁻²), by varying the combined relative flux density contribution of the secondary component(s) (up to 40% of the total) and also by varying the initial phases ψ of the secondary component(s) relative to the primary component, and to each other (between 0 and π). These variations reflect what we see in the FD distributions of our sample quasars (see ${\sigma }_{\mathrm{FD}}$ and C in Figure 10). Furthermore, each of these 6000 models is convolved with a range of ${\sigma }_{\mathrm{disp}}$ between 0 and 40 rad m⁻². The obtained DP parameters of those models are shown in the different panels of Figure 12. As with the real data, it was not always possible to fit ${\sigma }_{\mathrm{Burn}}$ (bottom left) and $\sigma {^{\prime} }_{\mathrm{Burn}}$ (bottom right) with physically meaningful values. The red lines in each panel show the behavior of the single component model. As a comparison, the distribution of the corresponding DP parameters in our observed quasar sample are represented by the histograms on the right-hand axis.

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As discussed above, the feature in the FD distribution that most drives depolarization is ${\sigma }_{\mathrm{disp}}$. Hence, for our purposes, a "good" DP parameter should be one that well traces ${\sigma }_{\mathrm{disp}}$. Figure 12 shows the relation of the previously defined DP parameters to ${\sigma }_{\mathrm{disp}}$ for the simple one component model (red line) and as a comparison for more complex models with two or three components (black lines). It shows how the appearance of additional components can bias DP and blur the relation between ${\sigma }_{\mathrm{disp}}$ and DP. We see that DP_Quart (top left) and DP_Half (top right) can be both increased and decreased by the presence of secondary components while DP${}_{\mathrm{Min}/\mathrm{Max}}$ (middle left) and DP${^{\prime} }_{\mathrm{Min}/\mathrm{Max}}$ (middle right) mostly tend to be increased.

Both ${\sigma }_{\mathrm{Burn}}$ (bottom left) and $\sigma {^{\prime} }_{\mathrm{Burn}}$ (bottom right) work surprisingly well in recovering an estimate of ${\sigma }_{\mathrm{disp}}$, despite the difficulties of fitting the Burn model due to the non-monotonic structure of ${\rm{\Pi }}({\lambda }^{2})$. For DP_Quart and DP_Half most of the objects in our sample are in that region where secondary components can strongly affect them. As opposed to DP_Quart or DP_Half, we see that for DP${}_{\mathrm{Min}/\mathrm{Max}}$ and DP${^{\prime} }_{\mathrm{Min}/\mathrm{Max}}$ a fair portion of the sample have values close to one, where the value is more robust against additional components.

This analysis, based on a simple toy model, therefore offers a possible explanation as to why we got different KS significances for the different DP parameters in Section 5.2. Recall that we saw the strongest correlations with the presence of intervening absorption with DP${}_{\mathrm{Min}/\mathrm{Max}}$ and DP${^{\prime} }_{\mathrm{Min}/\mathrm{Max}}$, and also, when measurable with ${\sigma }_{\mathrm{Burn}}$ and especially $\sigma {^{\prime} }_{\mathrm{Burn}}$. We would have seen this behavior if the presence of Mg ii absorption was associated with the fine structure ${\sigma }_{\mathrm{disp}}$ rather than the presence of the multiple components that dominate the visual impression of the FD distributions in Figures 1–3.

We therefore suggest that intervening material is primarily responsible for broadening the FD distribution $F(\phi )$ rather than the presence of the multiple discrete components. We will test this in the next Section 5.5 and provide a further argument in favor of this idea in the subsequent Section 5.6.

In the previous section, we postulated that intervening material, traced by the Mg ii absorption, affects the observed FD distribution by means of a convolution effect. ${\sigma }_{\mathrm{PC}}$, introduced in Section 3.3, is a parameter constructed to be insensitive to the appearance of multiple components.

In Figure 13, we compare the ${\sigma }_{\mathrm{PC}}$ to $\sigma {^{\prime} }_{\mathrm{Burn}}$. The ${\sigma }_{\mathrm{RMSF}}$ in the VLA data is around 17 rad m⁻² and is around 24 rad m⁻² for ATCA. Most objects have ${\sigma }_{\mathrm{PC}}$ smaller than that, i.e., those sources are barely resolved in ϕ space, and the values should be interpreted with some caution. Nevertheless, and despite the difficulties in obtaining fits to ${\rm{\Pi }}({\lambda }^{2})$ with the Burn model, we see a reasonable overall correlation between ${\sigma }_{\mathrm{PC}}$ and $\sigma {^{\prime} }_{\mathrm{Burn}}$.

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If we now construct our usual KS test between absorption and ${\sigma }_{\mathrm{PC}}$, i.e., a one-tailed test with the null hypothesis that objects with Mg ii absorption along the lines of sight are drawn from the same distribution of ${\sigma }_{\mathrm{PC}}$ as objects with clean lines of sight or that objects with clean lines of sight have enhanced ${\sigma }_{\mathrm{PC}}$, we can reject the null hypothesis with $p=3.5 \%$ (Figure 14). When re-scrambling objects with and without Mg ii absorption along the lines of sight as in Section 5.1 $p=3.5 \%$ or smaller happens in 1.9% of the realizations.

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To assess the significance of this result, it should be borne in mind that most of the objects were not resolved in ϕ space. It is noticeable that all of the quasars with ${\sigma }_{\mathrm{PC}}$ significantly larger than ${\sigma }_{\mathrm{RMSF}}$ have intervening Mg ii absorption.

When we use the fractional q and u for RM Synthesis, we obtain here an even more significant p-value, with $p=0.09 \%$. Sources with spectral indices different than zero, i.e., sources that vary along the observed frequency, will have associated variations also in the polarized flux density, i.e., in Q and U, and this will affect the FD distribution. Since these variations are normally gradual with frequency, these will rather affect parameters like ${\sigma }_{\mathrm{PC}}$ than adding new secondary components. This then would blur the KS Test with ${\sigma }_{\mathrm{PC}}$. Using fractional q and u is an attempt to take out this effect and indeed it yields a stronger result.

The strong association of both, depolarization and ${\sigma }_{\mathrm{PC}}$, with intervening material indicates that intrinsic effects within the sources do not contribute much to the fine structure of the Faraday Depth within each components, i.e., ${\sigma }_{\mathrm{disp}}$. This could be due to different properties of the magnetized plasma within sources compared to that in the intervening systems traced by Mg ii absorption. However, redshift effects should also be considered here. A given dispersion in the rest-frame $F(\phi )$ at some redshift z will produce an observed ${\sigma }_{\mathrm{disp}}$ that scales as ${(1+z)}^{-2}$. Since the quasars are generally at significantly higher redshifts than the intervening absorbing systems, this could explain why the observed ${\sigma }_{\mathrm{disp}}$ is dominated by the intervening systems.

We have argued above that the presence of multiple discrete components in the FD distributions $F(\phi )$ of our sources (to which the parameters we introduced in Section 5.3 were particularly sensitive) is not associated with the presence of intervening Mg ii absorption. In contrast, intervening material is clearly associated with the broadening of the FD distribution. This raises the possibility that the multiple components in $F(\phi )$ arise actually intrinsically to the radio sources.

That this is likely the case is indicated when we consider the initial phases ψ of the components. This is shown on Figures 1–3 (Column (6)). We always see that if a secondary component is present, then the initial phase of that secondary component is different than the initial phase of the primary component. In other words, the cusps in the radial ${\boldsymbol{F}}$ plots (in Column (6)), that correspond to the peaks in the $F(\phi )$ plots in Column (5), lie at different azimuthal angles representing the initial phase ψ.

An important point is that source components with different initial phase ψ must come from spatially distinct emission regions, either in the plane of the sky, or along the line of sight. This is because ψ represents the phase of emission before any Faraday Rotation. It is then quite reasonable to imagine that these different emission regions have associated with them different intrinsic ϕ. This would naturally produce the multiple ϕ components, each with distinct ϕ and ψ, in the overall FD distribution, as observed.

However, the argument above is not watertight: it could also be imagined that different source regions (with different ψ) lying behind different parts of an intervening system would also suffer vastly different amounts of Faraday Rotation. We could for instance imagine a chequer-board intervening screen with just two values of intervening ϕ, ${\phi }_{1}$, and ${\phi }_{2}$, which lies in front of a source in which ψ varies spatially. We would then observe a Faraday Depth with two separated peaks ${\phi }_{1}$ and ${\phi }_{2}$, each with a ψ that reflected the distribution of ψ of the source regions behind the ${\phi }_{1}$ and ${\phi }_{2}$ cells of the intervening screen. This would also produce two distinct components in $F(\phi )$ with different ψ, with all of the Faraday Rotation coming from the intervening system. However, in this case, we would expect to see a correlation between the presence of Mg ii and the parameters introduced earlier that are sensitive to the presence of these multiple components in the FD distribution. This is not seen, suggesting that indeed, the different ϕ of the multiple components originate intrinsically to the source.

Also, the fact that we do not see a single example in which two or more cusps have the same ψ, which would happen, e.g., if a chequer-board intervening screen as described before lies in front of a source with constant ψ, should be taken as an indication that distinct components in the FD distribution arise from intrinsic effects.

We conclude therefore that the gross structure complexity and multiple distinct components in $F(\phi )$ that are seen in many sources are produced by effects intrinsic to the sources, while intervening material introduces a small but pervasive broadening of the FD distribution. We explore further the implications of the latter effect in Section 6. Further exploration of the properties of the background sources from QU-fitting (O'Sullivan et al. 2012; Sun et al. 2015) and analysis of the shape of the ${\boldsymbol{F}}$ plots is beyond the scope of the current work, but we plan to develop this further in the future. The results of the QU-fitting will also be very interesting to be compared with ${\sigma }_{\mathrm{PC}}$.