# Chromatic optical polarization of BL Lac: while faint and bright

Elena Shablovinskaya, Eugene Malygin, Dmitry Oparin
Special astrophysical observatory of Russian Academy of Sciences, Nizhnij
Arkhyz, Karachai-Cherkessian Republic, 369167, Russia E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Due to the first results on astrophysical X-ray polarization provided by IXPE
observatory, the interest in wavelength-dependent synchrotron polarization of
BL Lac type objects increases. This paper presents the results of multi-band
optical observations of the well-known blazar named BL Lac ($z=0.069$) in
polarized light. It was shown that the object’s emission, regardless of its
phase of activity, is characterized by the intraday variability of brightness
and polarization with changes occurring on a time-scale of up to 1.5 hours
without any stable oscillation period. Polarimetric observations in the
different optical bands show that the degree and angle of polarization of the
blazar depend on the wavelength, and the maximum chromatism, as well as the
maximum observed polarization degree, was detected during the minimum
brightness state; during the flare state, the polarization chromatism changed
along with the flux gradient on the time-scale of an hour. Qualitatively, such
behaviour can be described by the shock-in-jet model, yet the chromatism
amplitude and its rapid changes differ significantly from the model
predictions and challenge the numerical calculations.

###### keywords:

BL Lacertae objects: individual: BL Lac – polarization – methods:
observational

††pubyear: 2022††pagerange: Chromatic optical polarization of BL Lac: while
faint and bright–A

##  1 Introduction

The different orientation of active galactic nuclei (AGN) relative to the
observer allows for studying various physical manifestations of the matter
accretion on to a supermassive black hole (SMBH) (Urry & Padovani, 1995;
Antonucci, 1993). BL Lac type objects, or blazars, are a special type of AGN,
where the relativistic jet is oriented so that one can reveal the processes
occurring inside the jet – from the regions very close to the SMBH in the
short-wave spectral range to parsecs and kiloparsecs scales in radio waves.

It has long been clear that determining the polarimetric properties of blazar
radiation, in particular the variability of polarization and its dependence on
wavelength, can help in studying the physical processes in the emitting plasma
and its dynamics, as well as the characteristics and structure of the magnetic
field in a jet (see e.g. Zhang, 2019, and references therein). A breakthrough
step there is associated with the launch of the IXPE satellite, which has
already allowed us to move much further in understanding the structure of the
jet. The observed differences between optical and X-ray polarization have
become an important argument in favour of the shock model in the blazar jet
(Liodakis et al., 2022).

| Date | JD | Device | Filter(s) | Duration | $\Delta t$
---|---|---|---|---|---|---
1 | 22/06/2020 | 022.981 | S |  $V$ \+ $I$ | 95 | 8
2 | 30/06/2020 | 030.946 | S | $V$ | 140 | 1.5
3 | 24/07/2020 | 054.920 | S | $V$ | 313 | 1
4 | 23/08/2020 | 084.941 | S |  $V$ \+ $I$ | 321 | 2
5 | 24/08/2020 | 085.839 | S |  $V$ \+ $I$ | 109 | 2
|  |  |  | $I$ | 172 | 1
6 | 24/10/2020 | 146.802 | M | CAMEL | 260 | 2
7 | 25/10/2020 | 147.785 | M | CAMEL | 265 | 2
8 | 28/06/2022 | 758.937 | M | CAMEL | 170 | 3.5
9 | 29/06/2022 | 759.908 | M | CAMEL | 236 | 5
10 | 30/06/2022 | 760.918 | M | CAMEL | 247 | 5
11 | 30/08/2022 | 821.904 | M | SED550 | 409 | 7
|  |  |  | \+ SED650 |  |
Table 1: Log of observations. The date is given in dd/mm/yyyy format, JD = JD
- 2459000.5. "S" is for the StoP device, and "M" is for the MAGIC focal
reducer. Duration and cadence $\Delta$t are given in minutes.

The processes observed in the optical range, in particular, rapid intraday
variability (IDV) of brightness and polarization in blazars, are described
qualitatively by the model of shock-waves in jet (Marscher et al., 2008;
Marscher, 2016), however, an accurate quantitative description has not yet
been obtained. In addition, the possible influence of other processes – both
the contribution of external AGN regions outside the jet and additional
processes inside it – remains uncertain. In the 1980s-90s there was a special
interest in the observation of polarization chromatism in the optical and
near-IR ranges (Bjornsson, 1985; Ballard et al., 1990, and etc.). There a wide
range of physical processes and models was proposed, which allows us to
explain in general terms the observed dependence of synchrotron polarization
on wavelength. In recent years, new models have also continued to appear (see
e.g. results in Marscher & Jorstad, 2021). Concerning the ongoing work of the
IXPE observatory, multi-wave observations of polarization within the optical
range seem an important addition and test for models aimed at describing the
polarization features in various spectral ranges.

To determine the characteristics of the optical frequency-dependent
polarization degree and angle, in 2020-2022 we conducted observations of the
well-known object BL Lac ($z=0.069$, RA = 22 02 43.3, Dec = +42 16 40, J2000),
the ancestor of the blazar objects class. Since the object showed historical
flares in 2020 and passed through a deep minimum phase in 2022, our interest
was to trace how the intraday variability of brightness and polarization
depends on the phase of the source activity, to assess whether the polarized
radiation of the object will be characterized by chromatism, and also to track
the change in these characteristics between epochs.

This paper is arranged as follows. Section 2 describes the observations of BL
Lac in 2020-2022 on the 1-m telescope of the SAO RAS; the results of these
observations are given in Section 3. Section 4 gives a detailed discussion on
the possible physical mechanisms of chromatism of BL Lac polarization. The
conclusion is given in Section 5.

Figure 1: Image of the BL Lac field obtained at Zeiss-1000 telescope using the
MAGIC device with a quadrupole Wollaston prism and a CAMEL filter. The
directions of the electric vector oscillation and the direction of the prism
dispersion are indicated for each of the four images. The image in the 0∘
direction shows an object and two stars – local standards. Blue and red
circles mark the "blue" and "red" images of the sources. Figure 2: CAMEL
filter transmission curve. The dashed line is the measured transmission curve,
the solid line is the transmission curve, convolved with the CCD sensitivity
curve. For observations, two peaks with CL=4600Å and CL=6300Å are used,
separated by $10^{\prime\prime}$ in the receiver plane due to the dispersion
of the quadrupole Wollaston prism. Artefacts of filter transmission in the
near-IR range form a parasitic image of sources on the frame, which does not
interfere with "blue" and "red" images.

##  2 Observations

Polarimetric observations of BL Lac were carried out in summer-autumn 2020 and
summer 2022 at the 1-m Zeiss-1000 telescope of SAO RAS (Komarov et al., 2020).
Two devices were used for observations: a photometer-polarimeter StoP
(Afanasiev et al., 2021) and a focal reducer MAGIC (Afanasiev et al., 2022a).
The general log of observations is given in Table 1. Data reduction included
standard steps of bias and flat-fielding subtraction.

In the polarimetric mode, the StoP device uses a double Wollaston prism of the
wedged design (Oliva, 1997). The image of the input mask with the size
$\sim$1${}^{\prime}\times$6′ was registered on the receiver in four directions
of the electric vector oscillation 0∘, 90∘, 45∘ and 135∘ separated so that
they do not overlap each other. In the case of MAGIC, a Wollaston prism of the
original design, the so-called quadrupole prism, proposed for the first time
by Geyer et al. (1993), is used. This prism also allows one to register four
images of the input mask, which in this case is a square of a
6.5${}^{\prime}\times$6.5′ size.

Double Wollaston prisms allow one to conduct observations in the so-called
one-shot polarimetry mode unaltered by rapid atmosphere flickering. Thus, the
Stokes parameters $I$, $Q$, $U$ are measured for each frame, calculated using
the formulae:

$\displaystyle I=I_{0}+I_{90}D_{\rm Q}+I_{45}+I_{135}D_{\rm U},$
$\displaystyle Q=\frac{I_{0}-I_{90}D_{\rm Q}}{I_{0}+I_{90}D_{\rm Q}},$
$\displaystyle U=\frac{I_{45}-I_{135}D_{\rm U}}{I_{45}+I_{135}D_{\rm U}},$

where $D_{\rm Q}$ and $D_{\rm U}$ are coefficients of polarization channel
transmission. Here and further in the text, $Q$ and $U$ are the Stokes
parameters normalized by intensity. $D_{\rm Q}$ and $D_{\rm U}$ are calculated
for each frame using the local standard in the field, which allows one not
only to take into account instrumental effects but also to correct atmospheric
variations during observations. For more details, see Afanasiev & Amirkhanyan
(2012). The polarization degree $P$ and the angle of the polarization plane
$\varphi$ then are equal to:

$\displaystyle{P}=\sqrt{Q^{2}+U^{2}},$
$\displaystyle{\varphi}=\frac{1}{2}\arctan Q/U.$

Note here that to transform the Stokes parameters to the celestial plane, one
should multiply the Stokes vector by the rotation matrix with a $-2\cdot$PA
angle, where PA is the positional angle of the device.

In the case of observations of BL Lac, we used the local standard #1 (see Fig.
1) at a distance of $\sim$5$"$ from the object as a reference star in all
epochs. The such closeness of the reference star to the source makes it
possible to minimize the possible residual effects of instrumental
polarization over the field, as well as to take into account the effects of
ISM polarization. However, since we are not taking the average polarization of
all the stars of the field, but only one star, the ISM polarization bias can
be introduced into the measured value, which may depend on the colour, but be
constant over time.

| Date |  Band |  <Mag> |  $F_{\rm var}$, % |  <$P$>, % |  <$\varphi$>, deg | <$\Delta$Mag> |
---|---|---|---|---|---|---|---|---
(1) | (2) |  (3) |  (4) |  (5) |  (6) |  (7) | (8) | (9)
1 | 22/06/2020 |  | $V$
---
$I$
| 14.42 $\pm$ 0.03
---
13.02 $\pm$ 0.02
| 9.4
---
8.3
| 7.7 $\pm$ 0.5
---
6.7 $\pm$ 0.2
| 48.6 $\pm$ 3.3
---
51.5 $\pm$ 1.9
1.4 | I
2 | 30/06/2020 |  $V$ |  13.95 $\pm$ 0.01 |  3.1 |  7.4 $\pm$ 0.5 |  160.3 $\pm$ 2.0 |  |
3 | 24/07/2020 |  $V$ |  14.23 $\pm$ 0.04 |  10.0 |  10.8 $\pm$ 0.9 |  75.8 $\pm$ 2.0 |  |
4 | 23/08/2020 |  | $V$
---
$I$
| 12.69 $\pm$ 0.03
---
11.39 $\pm$ 0.02
| 13.8
---
13.0
| 2.9 $\pm$ 1.2
---
3.3 $\pm$ 0.9
| 50.6 $\pm$ 12.7
---
48.4 $\pm$ 10.1
1.3 | II
5 | 24/08/2020 |  | $V$
---
$I$
| 12.74 $\pm$ 0.02
---
11.45 $\pm$ 0.02
| 9.5
---
8.4
| 8.9 $\pm$ 0.6
---
7.3 $\pm$ 0.3
| 120.6 $\pm$ 2.8
---
120.9 $\pm$ 2.2
1.3
|  |  $I$ |  11.51 $\pm$ 0.03 |  12.0 |  5.3 $\pm$ 1.5 |  124.0 $\pm$ 2.4 |
6 | 24/10/2020 |  | $B$
---
$R$
| 14.97 $\pm$ 0.02
---
12.96 $\pm$ 0.01
| 9.8
---
7.1
| 11.0 $\pm$ 1.2
---
11.3 $\pm$ 0.8
| 175.0 $\pm$ 2.6
---
172.1 $\pm$ 1.7
2.0 | III
7 | 25/10/2020 |  | $B$
---
$R$
| 14.71 $\pm$ 0.04
---
12.73 $\pm$ 0.03
| 15.2
---
13.2
| 7.2 $\pm$ 0.9
---
8.1 $\pm$ 0.9
| 170.2 $\pm$ 3.6
---
166.5 $\pm$ 3.1
2.0
8 | 28/06/2022 |  | $B$
---
$R$
| 16.31 $\pm$ 0.01
---
14.20 $\pm$ 0.01
| 3.5
---
3.5
| 26.1 $\pm$ 1.3
---
23.5 $\pm$ 0.5
| 16.3 $\pm$ 1.1
---
11.3 $\pm$ 0.5
2.1 | IV
9 | 29/06/2022 |  | $B$
---
$R$
| 16.25 $\pm$ 0.02
---
14.15 $\pm$ 0.01
| 4.9
---
6.5
| 26.4 $\pm$ 1.3
---
24.9 $\pm$ 0.5
| 18.9 $\pm$ 1.1
---
12.9 $\pm$ 0.6
2.1
10 | 30/06/2022 |  | $B$
---
$R$
| 16.35 $\pm$ 0.02
---
14.26 $\pm$ 0.01
| 5.1
---
4.1
| 25.5 $\pm$ 1.4
---
22.5 $\pm$ 0.8
| 17.8 $\pm$ 1.7
---
11.8 $\pm$ 0.8
2.1
11 | 30/08/2022 |  | $V$
---
$R$
| 14.22 $\pm$ 0.02
---
13.52 $\pm$ 0.02
| 6.3
---
6.8
| 12.0 $\pm$ 0.8
---
11.6 $\pm$ 0.7
| 1.8 $\pm$ 4.8
---
1.1 $\pm$ 4.2
0.7 | V
Table 2: Table of average values of brightness, amplitude of variability and
polarization of BL Lac depending on the epoch and photometric band: (1) epoch
number; (2) date of observations in the format dd/mm/yyyy; (3) photometric
band; (4) average magnitude; (5) amplitude of variability $F_{\rm var}$ in per
cent; (6) average degree of polarization <$P$> in per cent; (7) average angle
of polarization <$\varphi$> in degrees; (8) average difference of magnitudes
in two bands, characterizing the colour of the object; (9) the state of the
object activity (see Sec. 3).

In order for the effect of polarization chromatism to be detectable, we
selected observational bands that were as far apart in terms of wavelengths
from each other as possible. For observations on StoP, we chose the Johnson
system broadband filters $V$ and $I$. The observations in the same broad bands
with MAGIC quadrupole Wollaston prism are impossible since the prism has a
significant dispersion, much greater than the images formed by seeing (see
Afanasiev et al., 2022b, for more detailed characteristics of the prism).
Therefore, for all the nights of observations except the last one with the
MAGIC device, we used a so-called CAMEL filter produced by Chroma
Technology111https://www.chroma.com/ with two bandpasses. The filter
transmission curve is given in Fig. 2. The first band (CWL=4600Å, FWHM=330Å)
is oriented close to the maximum Johnson $B$-band and the second band
(CWL=6300Å, FWHM=460Å) is close to Johnson $R$-band, so further in the text we
will use the notation $B$ and $R$ for this observational mode222Note also that
CAMEL is medium-band (less than 500Å) filter with maximum transmission less
than 50%. Thus, the filter is suitable for observations of only bright
objects.. Also in Fig. 2 it can be seen that in the wavelength range $\sim$1
$\mu$m, filter transmission artifacts appear. Due to the use of a thick CCD
Andor iKon-L 936 BEX2-DD characterized by high sensitivity in the red and
near-IR ranges an artefact IR image of the sources is formed but does not
affect the "blue" and "red" images, so we neglect it. The dispersion of the
Wollaston prism separates the images in $B$ and $R$ bands at a distance of
$10^{\prime\prime}$. The distortion of star-shaped images due to the prism
dispersion does not exceed $3^{\prime\prime}$, which does not complicate
photometry. In the last epoch of observations (08/30/2022), we used MAGIC with
the usual wedged Wollaston prism from the StoP polarimeter device with a
narrow mask of $31"\times$9′ size. The observations were carried out with the
SED550 and SED650 medium-band filters from the SCORPIO-2
set333https://www.sao.ru/hq/lsfvo/devices/scorpio-2/filters_eng.html,
corresponding to $V$ and $R$ bands.

##  3 Results

In total, 11 epochs of BL Lac observations were obtained in 2020-2022. For 9
epochs, the object was observed in two different colours; 2 more epochs carry
additional information in only one colour. For each epoch, the light and
polarization curves are shown in Fig. 3 displaying the variations of
magnitude, the Stokes parameters $Q$ and $U$, polarization degree $P$ and
angle $\varphi$ with time. To correlate the variability of BL Lac with its
state in the context of a long-term trend, for each epoch in each
observational band, the average values of brightness, the amplitude of
variability, degree and angle of polarization and spectral indices are
calculated and given in Table 2.

Comparison of the obtained photometric data with open-access monitoring
data444For the analysis, we used open-access data from the American
Association of Variable Star Observers (AAVSO) website:
https://www.aavso.org/. allowed us to identify the states of the blazar
activity during the campaign:

  * I

– pre-flare: June-July, 2020;

  * II

– flare: August, 2020;

  * III

– post-flare: October, 2020;

  * IV

– minimum: June, 2022;

  * V

– post-minimum: August, 2022;

These states are also presented in Table 2.

Figure 3: The light and polarization curves for all epochs when the
observations were carried out in two bands. The panels correspond (from top to
bottom): magnitude in the "redder" band, the magnitude in the "blue" band,
Stokes parameters $Q$ and $U$ in per cent, the polarization degree $P$ in per
cent, the polarization angle $\varphi$ in degrees. The x-axis displays the
time in minutes from the beginning of observations in each epoch. The dots
colours correspond to the observational bands: filled blue dots denote the $B$
band, filled green dots denote the $V$ band, open red dots denote the $R$
band, and open purple dots denote the $I$ band.

###  3.1 Photometric IDV

As can be seen in Fig. 3, for all epochs, even when continuous observations
were carried out for only $\sim$1.5 hours, regardless of the object
brightness, the magnitude varied by at least 0.05 mag. For a more detailed
measurement of the amplitude of the object variability, we use the value
$F_{\rm var}$, introduced in the work (Heidt & Wagner, 1996) and often used by
other authors to characterize intraday brightness changes:

$F_{\rm var}=100\times\sqrt{({\rm Max}-{\rm Min})^{2}-2\sigma^{2}}\ \ (\%),$

where ${\rm Max}$ and ${\rm Min}$ are the maximum and minimum magnitudes of
the object during the night, respectively, and $\sigma$ is the average
measurement error. The obtained values of the variability amplitudes for each
night in each observational band are presented in Table 2. According to the
values of $F_{\rm var}$, the greatest amplitude of variability is observed
during the flare (II). On the contrary, the object shows weak variability
within the night during the period of the deep minimum (IV).

The BL Lac light curves in different epochs of observations demonstrate quasi-
sinusoidal changes. By analogy with the study of the variability of the blazar
S5 0716+714 (Shablovinskaya & Afanasiev, 2019; Afanasiev et al., 2021), a
wavelet analysis of the BL Lac light curves was carried out. However, in
contrast to the case of S5 0716+714, where in two time-separated epochs of
observations, the period of oscillation of the brightness was determined with
high accuracy as $\sim$1.5 hours, in for BL Lac it was not possible to
uniquely determine the period of variations for any epoch. This behaviour of
the object’s brightness may indicate either that this period is longer than
the duration of observations ranging from 1.5 to 7 hours, or non-stationary
processes causing changes in brightness with no harmonic oscillations. It is
important to note that Webb (2016) also did not reveal any repeatable time-
scales of BL Lac IDV during 31 cycles of micro-variability observations.

Moreover, for all epochs where observations were carried out in two bands, a
cross-correlation analysis of the light curves in two colours was carried out.
Since the time series are always even, the classical cross-correlation
function was used for the analysis. It has been shown that there is no time
delay between colours in total light together with $\sim$80-95% of
correlation. Multiwavelength optical observations presented by Kalita et al.
(2022) stated the same result for their observations in October and November
2020 in g-, r- and i-sdss filters.

Figure 4: $QU$-diagram for BL Lac. The epochs are indicated by the numbers as
in Tables 1 and 2. The colour change from light to dark corresponds to the
time from the beginning of the observations; the data were binned to the
uniform cadence of $\sim$8 minutes. The colours correspond to the
observational bands: blue is for the $B$ band, green – for the $V$ band,
orange-red – for the $R$ band and magenta – for the $I$ band. The inserts with
data in only the $R$ band have been added for clarity where the observational
data is densely populated.

###  3.2 Polarimetric IDV & QU-diagram

The light curves of the Stokes parameters $Q$ and $U$ and the polarization
degree $P$ and angle $\varphi$ are shown in Fig. 3 using colours matched to
the observational bands. The plots show that during all the observed epochs,
the polarization changes within the night, and the patterns of these changes
differ between the epochs. So, during the flare, we observe violent $P$ and
$\varphi$ variations of large amplitude, when the polarization angle was
changing dramatically by $\sim$50∘ in a few hours. Oppositely, during the
minimum state, the polarization of BL Lac had a stable angle and a slowly and
smoothly varying degree.

Figure 5: Variations of the polarization and flux of BL Lac during the flare
(08/23/2020). Upper left: $QU$-diagram in the $V$ (green) and $I$ (magenta)
bands. The colour indicates the time from the beginning (light) to the end
(dark) of observations. The grey arrow indicates the jet orientation obtained
from high-resolution radio data (Casadio et al., 2021). Bottom left: change of
the polarization angle difference $\varphi(V)-\varphi(I)$ in time. Upper
right: rotation of the differential polarization vector. The colour shows the
change in the observation time from blue to red. Bottom right: flux-flux
diagram, in mJy.

Furthermore, the mean polarization varies significantly between epochs. It is
important to note that during the flare, especially 08/23/2020, the degree of
polarization of the object drops to almost zero. This circumstance may
indicate possible turbulent processes in plasma (see more in Section 4 below).
The greatest polarization degree $P$ of the order of 25-30%, which is an
outstanding value even for blazars, is observed during the minimum period when
the brightness of the object was close to the faintest over the past 8 years
($R\approx 14.3$ mag). This coincides with the independent measurements of
polarization in $R$-band provided in (Middei et al., 2022). This is the
unexpected result of observations since it is usually assumed that
polarization flares should occur simultaneously with the maximum brightness
(see e.g. Itoh et al., 2016). However, the same anti-correlation was earlier
discovered in BL Lac behaviour (Gaur et al., 2014).

To demonstrate the polarization vector rotation, the Stokes parameters were
depicted on the $QU$-diagram in Fig. 4. The epoch numbers are indicated as in
Tables 1 and 2. The colour change from light to dark corresponds to the time
from the beginning of the observations; for uniformity, the data is binned
over $\sim$8 minutes. The colours correspond to the bands of observations:
blue is for the $B$-band, green is for the $V$-band, orange-red is for the
$R$-band, and magenta is for the $I$-band. For epochs 8-10, the diagram does
not show data in the $B$ band due to the large errors.

The $QU$-diagram shows that patterns reveal a diverse zoo of trajectories that
practically do not repeat. Only in two observational periods – during the
minimum (8-10) and post-flare (6-7) – the change in the position of the data
on the $QU$-plot is small; in other cases, especially during the flare period
(4-5), the position on the diagram changes significantly even between two
adjacent nights. Qualitatively, this behaviour is typical for a large number
of blazars, as can be seen on the $QU$-diagrams in, e.g., (Impey et al.,
2000). Some of the longest and most detailed polarimetric observations of BL
Lac were carried out in (Moore et al., 1982). The analysis of the rotation of
the polarization vector on the $QU$-plane in that work also showed the
randomness of these movements, which is why the rotations were called ’random
or drunkard’s walk’.

###  3.3 Polarimetric IDV in flare

It is worth considering the IDV of polarization vector in epoch 4 when the
maximum brightness of the object was registered over the entire monitoring
campaign. On the $QU$-plane in both observational bands (Fig. 5, upper left),
the polarization vector abruptly changes direction several times during the
night, forming a "triangular" trajectory. Changes in the direction of vector
rotation can be considered together with changes in the flux gradient. In the
flux-flux diagram (Fig. 5, bottom right) it can be seen that the radiation
intensity of the source first increases in both bands, reaches a maximum, and
then also synchronously decreases and begins to rise again. Note that this
behaviour is predicted in the case when the variability is determined by a
single component with a constant energy distribution (Hagen-Thorn & Marchenko,
1999).

Then, if we compare the moments of change in the polarization vector direction
on the $QU$-plane with the moments of change in the flux gradient, it turns
out that only during event 1 (see Fig. 5, about 70 minutes after the start of
the observations), the rotation of the vector occurs simultaneously with the
stop of the increase in the brightness of the source. When the vector rotates
the next time (about 160 minutes), this rotation is not reflected in the flux-
flux diagram in any way. In turn, the last event on the flux-flux diagram
(event 2, about 250 minutes) is represented on the $QU$-diagram by only minor
changes in the trajectory.

As previously shown in a number of papers, both long- and short-term changes
in the brightness and polarization of blazars can be described within the
simple geometric model of the motion of relativistic plasma in a helical
magnetic field. Indeed, if we consider the multiparametric model presented in
(Shablovinskaya & Afanasiev, 2019) for a large set of input parameters (e.g.
inclination angle, jet opening angle, etc.), then we can achieve similar
patterns of polarization change on the $QU$-plane. However, such a model, when
compared with real observational data, faces two difficulties. The first is
that in the case when polarization variations occur due to geometric effects,
the brightness of the source will change due to a change in the observed
Doppler factor (Butuzova, 2021). However, in this case, the changes in
polarization and brightness must occur synchronously, which contradicts
observations.

The second important feature found in the observations was that the position
of the polarization vector on the $QU$-plane is different in two different
colours. Showing the same pattern, the two trajectories are shifted and
rotated relative to each other by an angle of $\sim$20∘. The difference
between the two colours also evolves over the observations. Fig. 5, upper
right, shows the motion of the polarization vector on the $QU$-diagram as the
difference between two bands of observations, i.e. the polarization vector has
coordinates $\\{Q(V)-Q(I);U(V)-U(I)\\}$. It can be seen that this difference
varies over time. Accordingly, the difference in the polarization angles
$\varphi(V)-\varphi(I)$ (Fig. 5, bottom left) also changes. The
$\varphi(V)-\varphi(I)$ changes resemble a periodic oscillation, and the
extremes correspond to a change in the gradient on a flux-flux diagram.

##  4 Polarization chromatism

Figure 6: $QU$-diagram for the polarization difference between the "redder"
and "bluer" bands. Different epochs are indicated by different colours given
in the legend, the numbers of the epochs correspond to Table 1. The error bars
are indicated formally and correspond to the full spread of points during the
epoch.

The central result of our observations is that we found that for all epochs
there is a dependence of polarization on colour, and the effect is observed
both for the degree of polarization (i.e. frequency-dependent polarization
degree, FDP) and for the angle (frequency-dependent polarization angle, FDPA).
At the same time, the pattern of intraday changes in polarization in different
bands of observations is repeated, which is clearly seen by the example of the
polarization vector rotation during the flare (Section 3.3) and on similar
plots constructed for each of the epochs when observations were carried out in
two colours (see Section A). It is worth paying special attention to the fact
that the chromatism of polarization varies both from epoch to epoch, and even
within the night on the time-scale of hours, which is demonstrated on
differential $QU$-diagrams (see Section A).

To search for the characteristic features of frequency dependence in different
brightness states of BL Lac, Fig. 6 shows a $QU$-diagram of the polarization
difference between the conditionally "red" and "blue" bands. For the full set
of observations, it turned out that, depending on the phase of BL Lac
activity, colour polarization is located in different parts of the diagram. In
a relatively quiet state (pre-flare, post-flare, post-minimum), the
differential polarization is grouped in approximately one place, only slightly
different from zero. During the flare, the chromatism changes, and the red
polarized component dominates; the differential polarization passes into
another quadrant, almost the opposite of the quiet state. In general, for the
most of the epochs the polarization chromatism spreads between -2% and 2% both
for $Q$ and $U$ parameters. Optical multiwavelength observations of BL Lac
conducted by Imazawa et al. (2022) in 2020 (between 5th and 6th epochs of our
monitoring) and 2021 demonstrated the variations of chromatism in the same
range. However, according to our data, the exception is the period of the
minimum brightness of the object, where the chromatism is most sharply
manifested. Then not only the maximum polarization of the source is observed
(see Fig. 3), but also maximum chromatism, not typical for other periods.
Unlike the flare period, the blue component dominates the polarization during
the minimum state.

Since the blazar radiation shows all the signs of a synchrotron nature, the
observed polarization chromatism turns out to be a rather unexpected effect.
When we consider the radiation of relativistic electrons in a magnetic field
having a power-law energy distribution due to the effective acceleration of
any nature (it is usually customary to consider a shock acceleration or
magnetic reconnection), the polarization of synchrotron radiation does not
depend on the wavelength (Ginzburg & Syrovatskii, 1965). However, over the
years of blazars monitoring, chromatism of optical polarization has been
repeatedly observed. In the 1980s there was an opinion that the dependence of
polarization on wavelength is a rare phenomenon (e.g. Saikia & Salter, 1988).
However, already Sillanpää et al. (1993) found FDP and FDPA are regularly
observed in a large number of BL Lac type objects. The FDP of BL Lac itself
was shown, for example, in the work (Puschell & Stein, 1980), where in the
range 0.36-0.69$\mu$m the polarization smoothly changed depending on the
wavelength from 5% to 25%, reaching a maximum in the blue part of the
spectrum. In a number of later works (Mead et al., 1990; Sillanpää et al.,
1993; Tommasi et al., 2001) the FDP of BL Lac was repeatedly confirmed, and in
different epochs, both larger polarization in the red spectral range and vice
versa were detected. For all the observations available in these papers,
differential polarization diagrams similar to Fig. 6 were plotted. However, no
signs or features of polarization chromatism depending on the brightness of
the object were found.

Polarization chromatism is observed for different blazars both in the optical
and IR ranges (see Takalo et al., 1992, 1994; Efimov & Primak, 2006, and
references therein), and, for example, in radio band (see e.g. Park et al.,
2018; Kravchenko et al., 2017). Moreover, special attention is now focused on
the search for the dependence of polarization in the optical and X-ray ranges
(Liodakis et al., 2022). However, if the large difference between the
polarization in diverse spectral ranges can be explained by the influence of
different physical mechanisms, then the significant dependence of polarization
on wavelength and its variability within the optical band is still a subject
for discussion. Within the framework of this section, some of the most common
mechanisms proposed in the literature to explain the phenomena of FDP and FDPA
will be considered.

###  4.1 External processes?

In polarimetric observations, the observed polarization can have not only an
internal origin but also a significant contribution from external processes.
Among them, the most significant, especially for extragalactic objects, is
usually the polarization introduced by ISM. Due to the low galactic latitude
($b\approx-10^{\circ}$), the contribution of ISM polarization to BL Lac
observations is significant, and its accounting is carried out using
differential polarimetry using the local standard nearby. As shown above, ISM
polarization can be underestimated with this approach, which can introduce
bias depending on the wavelength. However, this effect will be small and
constant in time on the scale of observations, changing neither between epochs
nor within one night of observations. Similarly, we can reject the significant
contribution of wavelength-dependent scattering mechanisms (for example,
Rayleigh scattering) on the external structures of the AGN, since these
processes also do not explain the evolution of chromatism over time.

Also, it should be noted that even during the deep minimum of the object
brightness, we observed a high polarization value dominating in the blue part
band. This means that the contribution of ISM, scattering processes, as well
as depolarization due to the contribution of the host galaxy to the observed
optical radiation of BL Lac is negligible. Also, the mechanism of eclipsing
the optical jet regions of the blazar by an external cloud belonging to the
more distant parts of the jet during the minimum is not confirmed by
observations due to "bluer" polarization. Moreover, it is important to
highlight the potential impact of the accretion disc. Tommasi et al.
(2001)found for BL Lac in a non-extreme phase (13.7 mag in $V$) that the
polarization degree increases with the wavelength. It has been suggested that
the observed radiation may be significantly influenced by the accretion disc
as a thermal component. Similar behaviour was expected for the blazar in any
low-brightness state. However, the trend of increasing polarization in the red
range with a decrease of the brightness is not observed (Mead et al., 1990;
Sillanpää et al., 1993), and also completely contradicts the results of BL Lac
polarimetry at a minimum state in this work. Thus, even during the period of
minimum brightness, the observed optical radiation is still dominated by a
jet, and not an accretion disc. It could be shown since the polarization
models of the accretion disc do not predict a large degree of polarization: in
the case of a magnetized disc (see e.g. the model provided by Piotrovich et
al., 2021), the observed degree of polarization is $<$2%, and in the case of
Thomson scattering in an accretion disc wind (Beloborodov, 1998) the maximum
degree of polarization in extreme cases can reach 14%. Also, these models also
do not predict polarization variations.

We can also consider the possibility of the influence of Faraday rotation on
the optical radiation of a jet propagating in a magnetized medium. Earlier,
the Faraday effect has been repeatedly considered in the literature (e.g.
Sitko et al., 1984; de Diego et al., 1994, and references therein). Due to the
difference in the polarization angles in different spectral bands in our
observations, it is possible to estimate the value of RM. During the flare,
the difference $\Delta\varphi=|\varphi(I)-\varphi(V)|\approx 5^{\circ}=0.09$
radians. Then RM = 2.9 $\times$ 1011 rad m-2. Hence, $<n_{e}BD>\approx 6\times
10^{11}$, where $n_{e}$ is the electron density in cm-3, $B$ is the magnetic
field in Gauss, $D$ is the distance in pc. An estimate of the magnetic field
strength in an optical jet can be given assuming that the delay between two
observational bands is due to synchrotron losses of the high energy electrons
(Papadakis et al., 2003). During the monitoring of BL Lac given in this
article, no delay between different colours was detected in any of the epochs
for integral or polarized radiation. This gives us an upper limit on the
maximum possible delay, equal to the minimum observational cadence, i.e.
$\tau$ = 2 minutes = 120 sec. Following the formula from (Chiappetti et al.,
1999; Papadakis et al., 2003):

$B\delta^{1/3}\sim
300\left(\frac{1+z}{\nu_{I}}\right)^{1/3}\left[\frac{1-(\nu_{I}/\nu_{V})^{1/2}}{\tau}\right]^{2/3},$

where $\nu_{V}$ = 0.0055, $\nu_{I}$ = 0.0034 in units of 1017 Hz, we find
$B\delta^{1/3}\sim$ 29.5 G, or $B\sim$ 9.5-13.7 G, assuming that 10
$<\delta<30$ (Vermeulen & Cohen, 1994). The obtained result gives a lower
limit assuming given physical assumptions and is more than an order of
magnitude higher than the estimates of the magnetic field for BL Lac, obtained
earlier usually of the order of 0.5 G.

If we assume that the magnetic field strength in an optical jet is $B\approx
10$ G, then $<n_{e}D>\approx 6\times 10^{10}$. In other words, even if we
assume that the size of the region of magnetized matter through which optical
radiation passes is $D\sim 1$ kpc, then $n_{e}\sim 10^{7}$cm-3, which exceeds
estimates of the density of gas clouds in NLR ($n^{\rm NLR}_{e}\sim
10^{2}-10^{6}$cm-3). Based on these calculations, we can confidently conclude
that the Faraday effect does not significantly contribute to the observed
FDPA. Moreover, during the flare, $\Delta\varphi$ changes sign within the
night, which should correspond to a change in the orientation of the magnetic
field to the observer. Such significant changes in the orientation of the
magnetic field would also be difficult to describe within the framework of
modern jet models.

Thus, due to the evolution over time, the main contribution to the variability
of polarization and its dependence on wavelength is precisely the result of
internal physical processes in the optical jet.

###  4.2 Multi-component?

After detecting the dependence (and the variability of this dependence) of
polarization on the wavelength in blazars, the most common interpretation was
the model of multi-zone structure radiation. For example, in the work (Ballard
et al., 1990), among others, a simple model is given consisting of two
synchrotron components – polarized and unpolarized. Holmes et al. (1984) also
proposed a radiation model of two zones with different polarization and
spectral indices to explain the polarization characteristics of OJ 287.

Multi-component models are generally consistent with the commonly accepted
consideration of turbulent processes in a jet (see Marscher & Jorstad, 2021,
for a review and references therein). Simulations show that the presence of
turbulent cells in jet plasma can explain the rapid variability of blazars. A
variable number of cells increases or decreases the degree of ordering of the
magnetic field leading to polarization variability. In plasma, the dominant
contribution to radiation passes from one turbulent bubble to another. Due to
the different physical conditions, the multi-zone model could explain
variations in the chromatism of polarization within the night, especially
during the flare.

The numerical turbulent multi-zone model TEMZ was earlier presented in
(Marscher, 2014; Marscher et al., 2017). Marscher & Jorstad (2021) showed
simulation results for the case of a turbulent jet without a shock (see Figure
3 there). There, the greatest variability of polarization chromatism is
observed in the case of a 100% turbulent magnetic field without a regular
helical component. Even in this case, the minimum time to change the sign of
$dP/d\nu$ is $\sim$0.4 days, i.e. about 10 hours. This is equal to the
polarization frequency-dependency should not change significantly during the
night. In general, this is the case for most observational epochs (see Fig.
16), however, in the case of a flare when turbulent processes are considered
dominant, FDP and FDPA change on the scales of $\sim$1 hour.

It is worth noting that purely turbulent processes in plasma describe
qualitatively the observational data quite well. Moreover, the assumption of a
significant number of cells in a plasma with different physical
characteristics makes it possible to describe any variability by decomposing
it into a sufficient number of harmonics. At the same time, the application of
such an approach to real observational data faces a number of difficulties.
First of all, the cross-correlation analysis of multi-colour observations
shows that the variability in different colours is correlated in both total
and polarized light. Also, on the $QU$-plane, the rotation of the polarization
vector in different colours shows the same patterns up to rotation and shift,
regardless of the observation band. This is equivalent to the fact that the
radiating region behaves as one zone or, at least, the physical properties of
its individual components are interconnected. More importantly, in the results
of multi-colour polarization variations given by Marscher & Jorstad (2021) for
the case of plasma without a helical magnetic field, the model shows sharp
random jumps in the polarization angle. This behaviour is not confirmed either
in this article or by other authors. Polarimetric observations of blazars show
that switching and rotation of the polarization angle occur smoothly.

###  4.3 Synchrotron and acceleration?

As mentioned above, the polarization of pure synchrotron emission of a
homogeneous source does not depend on the wavelength. Then, to explain the
chromatism of polarization, models of the inhomogeneous distribution of the
magnetic field and electrons can be considered. In general, for the arbitrary
energy distribution of relativistic electrons in an arbitrary magnetic field,
the type of polarization dependence on frequency was obtained in (Bjornsson &
Blumenthal, 1982): $P(\nu)=\Pi(\nu)\cdot(\alpha(\nu)+1)/(\alpha(\nu)+5/3)$,
where $\Pi(\nu)$ contains information about the geometry and degree of
ordering of the magnetic field, and $\alpha$ is the spectral index. However,
such an approach to the description of polarization chromatism, as was
indicated in (Ballard et al., 1990), has no direct relation to the physical
model. Alternatively, Ballard et al. (1990) suggests, like Bjornsson (1985),
considering synchrotron radiation of cut-off electron distribution. In that
work, Figure 9 shows that in the presence of cut-off frequency, the dependence
of polarization on frequency has an asymmetric quasi-bell-shaped distribution
with a maximum degree of polarization at the cut-off frequency. In that work,
for a large data set of multi-wave polarization of a sample of blazars, the
data were approximated using this model, with the cut-off frequency as a free
parameter, which made it possible to describe observational data with any sign
of $dP/d\nu$. Thus, FDP can be described qualitatively and quantitatively by
the cut-off electrons synchrotron radiation model. However, this approach
excludes the existence of FDPA, which rises problems with describing BL Lac
data.

The assumption described by Ballard et al. (1990) is physically justified,
since the cut-off can be both a consequence of the presence of the maximum
frequency of electron emission (when energy losses are equal to the
acceleration energy gain) and, e.g., a passing shock wave. In general,
consideration of changes in the physical characteristics of plasma due to the
presence of shock processes in the jet (Marscher & Gear, 1985) is the most
popular approach today. The shock wave as an effective internal mechanism of
electron acceleration in jets is capable of simultaneously causing rapid flux
variability in a wide range of wavelengths and significant changes in the
electron energy distribution. In particular, the energy-stratified electron
distribution caused by the impact should generate strongly chromatic polarized
radiation, which is observed when comparing optical and X-ray polarimetric
data (Liodakis et al., 2022).

As shown by Imazawa et al. (2022), shock-in-jet model alone could not explain
all observed features, e.g. higher polarization degree in optical band than in
near-IR. Webb et al. (2017, 2021) provided polarimetric observations of the
sample of blazars in $VRI$ and $H$ bands aimed at testing of the two-zones
shock in jet model (Kirk et al., 1998). This model is based on the shock
excitement of turbulent cells, cooling then through synchrotron radiation.
This approach demonstrated excellent agreement with photometric IDV (Webb,
2016; Webb et al., 2021). However, polarimetric IDV as a critical test on the
original failed to show the predicted polarization behaviour, and the archival
data (Sasada et al., 2008; Bhatta et al., 2015) was too dense to draw any
conclusion. Using the TEMZ model for the case of a standing shock Marscher &
Jorstad (2021) calculated the polarization in the optical bands $B$, $R$, $I$.
These predictions are more suitable not only for the results obtained in this
paper but for all optical campaigns of BL Lac taken during 2020-2022. Unlike
the model without a shock, here, even in the absence of the helical component
of the magnetic field, the change in the polarization angle occurs smoothly,
which is generally consistent with observations. FDP and FDPA are also
observed regardless of the configuration of the magnetic field. Although TEMZ
results qualitatively agree well with the BL Lac polarimetry data, they still
cannot quantify them. First of all, due to the presence of a shock, the
chromatism of the optical polarization changes over a $\sim$2 times longer
time interval, which eliminates the variability of FDP on the scale of hours.
Moreover, the maximum chromatism between $B$ and $R$ bands is on the order of
$\sim$2% of the degree of polarization and does not exceed $\sim$10∘, which
also does not agree with the data.

Magnetic reconnection (Mizuno et al., 2011; Sironi et al., 2015; Petropoulou
et al., 2016) is considered to be an alternative acceleration mechanism
possible to explain the violent variability of the jet emission. Comparing
with the shock wave models, magnetic reconnection events also provide variable
polarization in all electromagnetic range yet the variability time-scales are
shorter and of the order of an hour or several hours (Zhang et al., 2022),
which coincides with known polarimetric IDV measurements. Moreover, according
to the present simulations (Zhang et al., 2020), magnetic reconnection leads
to the polarization chromatism of a detectable amplitude of at least several
per cents between optical and near-IR bands. Qualitatively this approach
agrees with the observational data. Further, much more exact approximation of
available polarimetric data via reconnection simulations should be worked out.

##  5 Conclusion

A significant number of epochs of optical polarimetric observations of BL Lac
obtained during 2020-2022 allowed not only determining the rapid changes in
the brightness and polarization of BL Lac but also correlating these changes
with long-term variability.

1\. In any state of activity, BL Lac demonstrates photometric IDV, even when
the object is in the minimum phase. Cross-correlation analysis showed no time
delay between the different colours in which observations were carried out
quasi-simultaneously during each epoch. This allows one to put the upper limit
on the delay equal to the minimum cadence time of 2 min. A wavelet analysis of
the observational data was also carried out, but periodic changes in the
integral and polarized fluxes could not be identified.

2\. Polarimetric IDV is observed in any state of BL Lac activity. The
$QU$-plane showed that the polarization state does not have a priority
location or a phase-dependent position on the diagram. During the period of
activity, both the degree and the angle of polarization change, and the
polarization degree is as low as $\sim$3%. During the minimum state, there are
practically no changes in the polarization angle, the polarization degree
varies slightly and smoothly, but is of $\sim$25-30%. Qualitatively, this
polarization behaviour is consistent with the dominance of turbulent processes
in the plasma during the flare and, conversely, a higher degree of
organization of the magnetic field during the minimum brightness state of the
blazar.

3\. All epochs of BL Lac observations are characterized by the presence of
chromatism of optical polarization. The variability of this chromatism between
epochs and inside the night cannot be explained by any external mechanisms or
the influence of components outside the jet. Thus, the variable chromatism of
polarization indicates internal mechanisms in the optical jet capable of
significantly changing the physical parameters of the emitting plasma on
scales up to 1.5 hours. Many models of inhomogeneous (broken) electron
distribution, including shock-in-jet and magnetic reconnection models, predict
the presence of chromatism of optical polarization. However, none of the
models discussed above can fully describe the obtained observational data
quantitatively. The main challenges for the models are (i) the BL Lac flare
when the sign of $dP/d\nu$ changed on the scales of several hours during the
night, and the polarization degree was quite low, and (ii) the period of a
deep minimum of the object brightness, when the jet still dominates, and the
polarization degree turns out to be maximal for the entire set of observations
and the most wavelength-dependent.

## Acknowledgements

The work was performed as part of the SAO RAS government contract approved by
the Ministry of Science and Higher Education of the Russian Federation.
Observations with the SAO RAS telescopes are supported by the Ministry of
Science and Higher Education of the Russian Federation.

## Data Availability

The reduced data underlying this article is available upon the request to the
corresponding author 1 yr after the publication of this paper.

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##  Appendix A BL Lac light curves of polarization and brightness

The section below presents additional figures demonstrating the observational
data obtained in 2020-2022. In Fig. 7 the light and polarization curves are
presented, similar to Fig. 3, for epochs when intraday polarimetric
observations were carried out only in the $V$ filter. Fig. 8-15 are similar to
Fig. 5 for all other epochs not given in the main text of the paper. In more
detail, the characteristics of the evolution of polarization chromatism during
observational epochs as a function of time, as well as the evolution of the
colour index from the degree of polarization, are given in Fig. 16.

Figure 7: The light and polarization curves for the epochs in which
observations were carried out in one band. The panels correspond (from top to
bottom) to: the magnitude in the $V$ band, the Stokes parameter $Q$ and $U$ in
per cent, the degree of polarization $P$ in per cent, and the polarization
angle $\varphi$ in degrees. The x-axis displays the time in minutes from the
beginning of observations in each epoch.  Figure 8: Variations of the
polarization and flux of BL Lac at 22/06/2020. Upper left: $QU$-diagram in the
$V$ (green) and $I$ (magenta) bands. The colour indicates the time from the
beginning (light) to the end (dark) of observations. The grey arrow indicates
the jet orientation. Bottom left: change of the polarization angle difference
$\varphi(V)-\varphi(I)$ in time. Upper right: rotation of the differential
polarization vector. The colour shows the change in the observation time from
blue to red. Bottom right: flux-flux diagram, in mJy. Figure 9: Similarly to
Fig. 8 for the $V$ (green) and $I$ (magenta) bands in the epoch 24/08/2020.
Figure 10: Similarly to Fig. 8 for the $B$ (blue) and $R$ (orange-red) bands
in the epoch 24/10/2020. Figure 11: Similarly to Fig. 8 for the $B$ (blue) and
$R$ (orange-red) bands in the epoch 28/06/2022. Figure 12: Similarly to Fig. 8
for the $B$ (blue) and $R$ (orange-red) bands in the epoch 30/06/2022. Figure
13: Similarly to Fig. 8 for the $B$ (blue) and $R$ (orange-red) bands in the
epoch 25/10/2020. Figure 14: Similarly to Fig. 8 for the $B$ (blue) and $R$
(orange-red) bands in the epoch 29/06/2022. Figure 15: Similarly to Fig. 8 for
the $V$ (green) and $R$ (orange-red) bands in the epoch 30/08/2022.

Figure 16: Light curves showing the difference in brightness and polarization
between two colours. The panels correspond (from top to bottom): the
difference in the polarization degree $P({\rm bluer})-P({\rm redder})$, the
difference in the polarization angle $\varphi({\rm bluer})-\varphi({\rm
redder})$, the difference in the magnitudes between the bands depending on the
time in minutes, the difference in magnitudes between the bands depending on
the polarization degree in the "bluer" band. Linear approximations of the data
are given in grey lines. In the panels where indicated the parameter ${\rm a}$
describes the slope of the approximating line.