-
Notifications
You must be signed in to change notification settings - Fork 3
/
thesis-single-file.tex
4185 lines (3531 loc) · 356 KB
/
thesis-single-file.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
\documentclass[goettingen,print]{thesis}
\usepackage[linktocpage]{hyperref}
%\usepackage{microtype}
%\usepackage[protrusion={true, compatibility}, expansion=false]{microtype}
\usepackage{txfonts}
\usepackage{natbib}
\usepackage{fancybox}
\usepackage{multirow}
\usepackage{mathrsfs}
\usepackage{wrapfig}
\usepackage{subfig}
\usepackage{setspace}
\usepackage{verbatim}
\usepackage[figuresright]{rotating}
\hypersetup{colorlinks=true, linkcolor=blue, citecolor=red, bookmarksopen=true, bookmarksnumbered=true, pdfstartpage={1}, pdftitle = {Observations, analysis and interpretation with non-LTE of chromospheric structures of the Sun}, pdfsubject = {PhD thesis }, pdfauthor = {Bruno S\'anchez-Andrade Nu\~no} , pdfkeywords = {}, pdfproducer = {LaTeX with hyperref running on OSX}}
\title{Observations, analysis and interpretation with non-LTE of chromospheric structures of the Sun}
\author{Bruno S\'anchez-Andrade Nu\~no}
\town{Oviedo, Asturias, Spanien}
\refereea{Prof. Dr. F. Kneer}
\refereeb{Prof. Dr. W. Kollatschny}
\submitteddate{Januar}
\submittedyear{2008}
\examinationdate{15 Februar, 2008}
\publicationyear{2008}
\isbn{ 978-3-936586-81-7 }
%\def\thechapter{\Roman{chapter}}
\def\thesubsubsection {\thesubsection .\alph{subsubsection}}
%\doublespacing
\begin{document}
\setstretch{1}
\maketitle
\include{firstpage}
\tableofcontents
\chapter*{Summary\markboth{Summary}{Summary}}
\addcontentsline{toc}{chapter}{Summary}
%? on .tex files to see comments. Consistency. Use THIS: Chapter Section broadband diffraction $\times$ Sun timescales center
%Follow ToC
%1 y 2 intro and spectral lines
\begin{comment}
{\textbf{Remove this paragraph?} The first scientific work about the chromosphere of the Sun can be trace back 150 years. Since then it has remained an exciting field of research. The chromosphere is best seen in spectral lines with strong absorption like H$\alpha$ and \ion{Ca}{II} K 3934 \AA\, or with special conditions of formation as the \ion{He}{i} 10830\, \AA\,line. Its wealth of dynamic phenomena includes an overall field of short fibrils covering much of the Sun, spicule ejecta seen outside the solar limb, bright plage regions associated with magnetic activity, dark filaments across the disc, prominences standing high above the photosphere or complex structures outlining the magnetic field in the solar atmosphere. The chromosphere owes its existence ti highly dynamic processes which occur in a wide range of temporal scales, a consequence of the special physical conditions in this solar atmospheric layer. In the much denser photosphere below the chromosphere the dynamics are dominated by the plasma motion, while higher up in the corona the gas pressure has decreased and the magnetic fields and their variation determine the plasma motions. In the chromosphere, the physical parameters are found to be between the two regimens.}
\end{comment}
%3 Observations
This thesis is based on observations performed at the \emph{Vacuum Tower Telescope} at the \emph{Observatorio del Teide}, Tenerife, Canary Islands. We have used an infrared spectropolarimeter (Tenerife Infrared Polarimeter -- TIP) and a Fabry-Perot spectrometer (``G\"ottingen'' Fabry-Perot Interferometer -- G-FPI). Observations were obtained during several campaigns from 2004 to 2006. We have applied methods to reduce the atmospheric distortions both during the observations and afterwards in the case of the G-FPI data using image processing techniques.
%4 HR in the chr.
We have studied chromospheric dynamics inside the solar disc. The G-FPI provides means to obtain very high spatial, spectral and temporal resolution. We observe at several wavelengths across the H$\alpha$ line. With different post-processing techniques, we achieve spatial resolutions better than $0\farcs5$. We present results from the comparison of the different image reconstruction methods. A time series of 55~min duration was taken from AR\,10875 at $\vartheta\approx36\degr$. From the wealth of structures we selected areas of interest to further study in detail some ongoing processes. We apply non-LTE inversion techniques to infer physical properties of a recurrent surge. We have studied the occurrence of simultaneous sympathetic mini-flares. Using temporal frequency filtering on the time series we observe waves along fibrils. We study the implications of their interpretations as wave solutions from a linear approximation of magneto-hydrodynamics. We conlude that a linear theory of wave propagation in straight magnetic flux tubes is not sufficient.
%5 infrared polar.
Furthermore, emission above the solar limb is investigated. Using infrared spectroscopic measurements in the \ion{He}{i} 10830\,\AA\ multiplet
we have studied the spicules outside solar disc. The analysis shows the variation of the off-limb emission profiles as a function of the distance to the visible solar limb. The
ratio between the intensities of the blue and the red components of this triplet $({\cal R}=I_{\rm blue}/I_{\rm red})$ is an observational signature of the optical thickness along the light path, which is related to the intensity of the coronal irradiation. The observable ${\cal R}$ as a function of the distance to the visible limb is given. We have compared the observational ${\cal R}$ with the intensity ratio obtained from \citet{Centeno06}, using detailed radiative transfer calculations in semi-empirical models of the solar atmosphere assuming spherical geometry
%FRANZ we? add cite to rebe
. The agreement is purely qualitative. We argue that this is a consequence of the limited extension of current models. With the observational results as constraints, future models should be extended outwards to reproduce our observations. To complete our analysis of spicules we report observational properties from high-resolution filtergrams in the H$\alpha$ spectral line taken with the G-FPI. We find that spicules can reach heights of 8 Mm above the limb. We show that spicules outside the limb continue as dark fibrils inside the disc.
\nopagebreak
One and a half centuries after the hand-drawings by Secchi, the chromosphere is still a source of unforeseen and exciting new discoveries.
\include{Introduction}
\include{Spectropolarimetry}
\include{Observations}
\include{HR}
\include{Spicules}
\include{Conclusions}
%\appendix
%\chapter{Apendix 1}
\bibliographystyle{thesis}
\bibliography{thesis}
\chapter*{Publications\markboth{Publications}{Publications}}
\addcontentsline{toc}{chapter}{Publications}
%\thispagestyle{empty}
\begin{enumerate}
\item[] \emph{Refereed papers}
\item
\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, R.~{Centeno}, K.~G. {Puschmann},
J.~{Trujillo Bueno}, J.~{Blanco Rodr{\'{\i}}guez}, and F.~{Kneer}.
\newblock {Spicule emission profiles observed in \ion{He}{i}~10830~\AA}.
\newblock {\em \aap}, 472:L51--L54, Sept. 2007.
\item
\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, N.~Bello Gonz\'alez, J.~{Blanco Rodr{\'{\i}}guez}, F.~{Kneer}, and K.~G. {Puschmann}.
%\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, N.~Bello Gonz\'alez, J.~{Blanco Rodr{\'{\i}}guez}, and F.~{Kneer}.
\newblock {Fast events and waves in an active region of the Sun
observed in H$\alpha$ with high spatial resolution}.
\newblock {\em \aap}, in press, Dec. 2007.
\item
J.~{Blanco Rodr{\'{\i}}guez}, O.~V. {Okunev}, K.~G. {Puschmann}, F.~{Kneer},
and \textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}.
\newblock {On the properties of faculae at the poles of the Sun}.
\newblock {\em \aap}, 474:251--259, Oct. 2007.
\item[] \emph{Conference contributions}
\item
\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, R.~{Centeno}, K.~G. {Puschmann},
J.~{Trujillo Bueno}, and F.~{Kneer}.
\newblock {Off-limb spectroscopy of the He I 10830 {\AA} multiplet:
observations vs. modelling}.
\newblock In F.~{Kneer}, K.~G. {Puschmann}, and A.~D. {Wittmann}, editors, {\em
Modern solar facilities - advanced solar science},
pages 177--180, 2007.
\item
\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, K.~G. {Puschmann}, and F.~{Kneer}.
\newblock {Observations of a flaring active region in H$\alpha$}.
\newblock In F.~{Kneer}, K.~G. {Puschmann}, and A.~D. {Wittmann}, editors, {\em
Modern solar facilities - advanced solar science},
pages 273--276, 2007.
\item
\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, K.~G. {Puschmann}, M.~{S{\'a}nchez Cuberes},
J.~{Blanco Rodr{\'{\i}}guez}, and F.~{Kneer}.
\newblock {Analysis of a Wide Chromospheric Active Region}.
\newblock In D.~E. {Innes}, A.~{Lagg}, S.~A. {Solanki}, and D.~Danesy, editors, {\em
Chromospheric and Coronal Magnetic Fields}, volume 596 of {\em ESA Special
Publication}, Nov. 2005.
\item
\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, K.~G. {Puschmann}, M.~{S{\'a}nchez Cuberes},
J.~{Blanco Rodr{\'{\i}}guez}, and F.~{Kneer}.
\newblock {Chromospheric Dynamics of a Solar Active Region}.
\newblock In {\em The Dynamic Sun: Challenges for Theory and Observations},
volume 600 of {\em ESA Special Publication}, Dec. 2005.
\item
\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}.
\newblock {Study case: Solar Science Communication}.
\newblock In L.~Lindberg Christensen, and M.~Zoulias, editors, {\em
Communicating Astronomy with the Public 2007}. An IAU/Nat. Obs. of Athens/Eugenides Foundation Conference, {\em } Oct. 2007.
\item
J.~{Blanco Rodr{\'{\i}}guez}, \textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, K.~G.
{Puschmann}, and F.~{Kneer}.
\newblock {Study of Polar Faculae}.
\newblock In {\em The Dynamic Sun: Challenges for Theory and Observations},
volume 600 of {\em ESA Special Publication}, Dec. 2005.
\item
J.~{Blanco Rodr{\'{\i}}guez}, \textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, K.~G.
{Puschmann}, and F.~{Kneer}.
\newblock {Study of Polar Faculae}.
\newblock In D.~E. {Innes}, A.~{Lagg}, S.~A. {Solanki}, and D.~Danesy, editors, {\em
Chromospheric and Coronal Magnetic Fields}, volume 596 of {\em ESA Special
Publication}, Nov. 2005.
\item
F.~{Kneer}, K.~G. {Puschmann}, J.~{Blanco Rodr{\'{\i}}guez},
\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, and A.~D. {Wittmann}.
\newblock {Magnetic Structures on the Sun: Observations with the New
''G{\"O}TTINGEN'' Two-Dimensional Spectrometer on Tenerife}.
\newblock In D.~E. {Innes}, A.~{Lagg}, and S.~A. {Solanki}, editors, {\em
Chromospheric and Coronal Magnetic Fields}, volume 596 of {\em ESA Special
Publication}, Nov. 2005.
\item
L.~{Valdivielso Casas}, N.~{Bello Gonz{\'a}lez}, K.~G. {Puschmann},
\textbf{B.~{S{\'a}nchez-Andrade Nu{\~n}o}}, and F.~{Kneer}.
\newblock {Analysis of Polarimetric Sunspot Data from Tesos/vtt/tenerife}.
\newblock In D.~E. {Innes}, A.~{Lagg}, S.~A. {Solanki}, and D.~Danesy, editors, {\em
Chromospheric and Coronal Magnetic Fields}, volume 596 of {\em ESA Special
Publication}, Nov. 2005.
\item
C.~Denker, A.P.~Verdoni, F.~W\"oger, A.~Tritschler, T.R.~Rimmele, F.~Kneer,
K.~Reardon, \textbf{B.~S\'anchez-Andrade Nu\~no}, I.~Dom\'inguez Cerde\~na and K.G.~Puschmann
\newblock {Speckle Interferometry of Solar Adaptive Optics Imagery}
\newblock {DFG-NSF Astrophysics Research Conference ``Advanced Photonics in Application to Astrophysical Problems'', June 2007}%, Washington, DC }
% \thispagestyle{empty}
\end{enumerate}
\include{acknowledgments}
%\chapter*{Acknowledgements\markboth{Acknowledgements}{Acknowledgements}}\addcontentsline{toc}{chapter}{Acknowledgements}
\chapter*{Lebenslauf\markboth{Lebenslauf}{Lebenslauf}}
\addcontentsline{toc}{chapter}{Lebenslauf}
%\thispagestyle{empty}
\begin{table}[htdp]
\begin{tabular}{p{.23\linewidth}lp{.35\linewidth}}
\textbf{Name:} & \multicolumn{2}{l}{Bruno S\'anchez-Andrade Nu\~no} \\
\\
\textbf{Geburtsdatum:} & \multicolumn{2}{l}{6. Mai 1981} \\
\\
\textbf{Geburstort:} & \multicolumn{2}{l}{Oviedo, Asturias, Spanien} \\
\\
\textbf{Familienstand:} & \multicolumn{2}{l}{Ledig} \\
\\
\textbf{Eltern:} & \multicolumn{2}{l}{Julio Miguel S\'anchez-Andrade Fern\'andez} \\
& \multicolumn{2}{l}{Concepci\'on Nu\~no L\'opez } \\
\\
\\
\textbf{Staatsangeh\"origkeit:} & \multicolumn{2}{l}{Spanisch} \\
\\
\\
\textbf{Schulbildung:} & September 1987 - Juni 1995 & Grundschule am \emph{Colegio P\'ublico ``R\'io Piles''} in {Gij\'on}, Asturias \\
& September 1995 - Juni 1999 & Weiter f\"uhrende Schule am \emph{I. B. ``R\'io Piles''} in {Gij\'on}, Asturias
\\
\textbf{Studium:} & September 1999 - September 2003 & Physikalische Fakult\"at der Universit\"at Oviedo, Asturias \\
& September 2003 - September 2004 & Physikalische Fakult\"at der Universit\"at La Laguna, Teneriffa (Fachrichtung Astrophysik) \\
\\
\textbf{Promotion:} & Januar 2005 - Januar 2008 & Promotion an der Universit\"ats-Sternwarte G\"ottingen (seit Juni 2005 Institut f\"ur Astrophysik G\"ottingen, IAG) \\
& Januar 2005 - Januar 2008 & Stipendium des Max-Planck-Instituts f\"ur Sonnensystemforschung \\
\end{tabular}
\end{table}
\end{document}
\chapter{Introduction}
This thesis deals with the chromosphere of the Sun. To give some insight to the readers which are not familiar with the topics of this work we introduce in Section \ref{intro:sun} the main characteristics of the Sun with a short general description. This will elucidate the position of the chromosphere in the solar structure and its role for the outer solar atmosphere. In the subsequent Section \ref{intro:chromo}, those aspects of the chromosphere which are treated in the present work are specified. Finally Section \ref{ref:out} indicates the structure of this thesis work.
\section{The Sun\label{intro:sun}}
\begin{flushright}
\emph{It is just a ball of burning gas\\ \dots right?\\
\vspace{1cm}}
\end{flushright}
The Sun is the central object of the Solar System, which also contains planets and many other bodies such as planetoids (small planets), comets, meteoroids and dust particles. However, the Sun on its own harbors 99.8\% of the total mass of the system, so all other objects orbit around it.
The Sun itself orbits the center of our Galaxy, the \emph{Milky Way}, with a speed of $217$ km/s. The period of revolution is $\sim230$ million years (the last time the Sun was on this part of the Galaxy was the time the Dinosaurs appeared). Compared to the population of stars in our galaxy, the Sun is a middle-aged, middle-sized, common type star. In astrophysicist's language it is of spectral type \emph{G2} and of luminosity class \emph{V}, located on the main sequence of stars in the Hertzsprung-Russell diagram. According to our understandings derived from models, it has been on the main sequence for $5\,000$ million years and it will remain there for another $5\,000$ million years before starting the giant phase.
The Sun is the closest star to us, the next one being $250\, 000$ times further away, but still light from the Sun's surface takes around 8 minutes to reach the Earth. It is the only star from where we get enough energy to study its spectrum in great detail and with short temporal cadence. With indirect methods, we can produce images of the surface structuring on other nearby starts. But on the Sun, with current telescopes and techniques, we resolve structures down to 100 km size on its surface, which represents approximately the resolution limit in this thesis work. We can also investigate the structure of its atmosphere and the effects of its magnetism. Actually, we are embedded in the solar wind that has its origin in the outer solar atmosphere, the corona of the Sun. Thus, we can make \emph{in-situ} measurements. With special techniques and models, we can reconstruct the properties of its interior.
% franz I refer to helioseismology, for example.
\pagebreak
\begin{wrapfigure}[14]{r}{0.5\textwidth}
\vspace{0.5cm}
\begin{center}
\includegraphics[width=0.5\textwidth]{../figures/sol.jpg}
\caption{The apparent size of the Sun on the sky is $\sim32 ' $, a little bit larger than one half degree. }
\label{fig:foto:sol}
\end{center}
\end{wrapfigure}
The Sun is the most brilliant object in the sky, 12 orders of magnitude brighter than the second brightest object, the full Moon, which actually only reflects the sunlight. Its light warms the surface of the Earth and is used by plants to grow. Its radiation is the input for the climate. The solar wind separates us from the interstellar medium. The magnetism of the Sun protects us from cosmic high-energy radiation and it influences the climate on Earth. Violent events in the solar ultraviolet radiation and the solar wind can also disrupt radio communications.
The Sun possesses a complex structure. Essentially, it can be described as a giant conglomerate of Hydrogen and Helium ($\sim74$\% and $\sim24$\% of the mass, respectively) and traces of many other chemical elements. Due to its big mass the self-gravitation keeps the structure as a sphere. From the weight of the outer spherical gas shells the pressure increases towards the center of the sphere. During the gravitational contraction of the pre-solar nebula towards its center, i.e. when forming a protostar, the gas has heated up by converting potential energy into thermal (kinetic) energy. This produces, together with a high gas density, a high pressure, which prevents the sphere from collapsing further inwards. Eventually, near the center, temperature and pressure are high enough to ignite nuclear reactions.
\subsubsection*{Structure}
At the core of the Sun the density and temperature (of the order of 13 million Kelvin, or $13 \cdot 10^{6}$ K) are high enough to fuse hydrogen and burn it into helium. This process also produces energy in the form of high-energy photons. This continuous, long-lasting energy output from the nuclear reactions keeps the core of the Sun at high temperature to sustain the gravitational load from the outer gas shells. Due to the high density, the photons are continuously absorbed and re-emitted by nearby ions, and in this way the big energy output is slowly \emph{radiated} outwards, while, towards the surface of the Sun, the density decreases exponentially, along with the temperature. Photons reaching today the Earth's surface were typically generated on the early times of \emph{Homo Sapiens}, as the typical travel time is $\sim170\,000$ years \citep{1992ApJ...401..759M}.
At a distance from the center of approximately 70\% of the solar radius, the radiation process is not efficient enough to transport the huge amount of energy produced in the core. There the gas is heated up, and expands, it becomes buoyant and rises. This creates \emph{convection} cells in which hot material is driven up by buoyancy while cool gas sinks to the bottom of the cells, where it is heated again. These gas flows transport the energy to the outer part of the Sun, where the temperature is measured to be $\sim 5\,700\,$K and the density is low enough that the photons can escape without much further absorption. The outer region from where we receive most of the optical photons can be called the surface of the Sun, although it is not a layer in the solid state. It is called the \emph{photosphere} (sphere of light). Most of the photons we receive come from this layer are in the \emph{visible} part of the spectrum: light. This is why Nature favored in the late evolution process the development of vision instruments that are more sensitive %
in the spectral region in which most emission from the Sun occurs.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{../figures/mosaicb.png}
\caption{High resolution image of the ``surface'' (photosphere) of the Sun with a resolution of $\sim140$ km. Granules are seen all around the photosphere outside the dark areas. They form the uppermost layers of the convection zone, in which the energy is transported from deep down outwards via gas motions. At the top, the gas cools down by radiating photons into space. Localized strong magnetic fields can also emerge and are seen as dark areas, the sunspots, which are a consequence of the less efficient energy transport.}
\label{fig:photosphere}
\end{center}
\end{figure}
Further out of this layer the atmosphere of the Sun extends radially, with decreasing density. In this outer part, with its low density, magnetic fields rooted inside the Sun cease to be pushed around by gas flows. This transition occurs together with a still not completely understood increase of temperature up to several million degrees. Therefore, there must be a layer with a minimum temperature. Standard average models place it at a height of about 500 km with a temperature of about 4000 K, which is low enough to allow the formation of molecules like CO or water vapor. Beyond this layer the temperature rises outward. Again in standard models, the layer following the temperature minimum has an extent of about 1\,500 km and its temperature rises to 8\,000 -- 10\,000 K. This layer is called the \emph{chromosphere}. The present work deals with some of its properties. Outside the chromosphere, the temperature rises abruptly within the \emph{transition region}. The outermost part of the atmosphere, called \emph{corona}, drives a permanent outwards flow of particles moving along the magnetic field lines. This \emph{solar wind} extends to $100\,000$ times the solar radius, far beyond Pluto's orbit, to the outer border of our Solar System, the \emph{heliopause}. There the interaction with the interstellar medium creates a shock front, which is being measured these years by the Voyager 1 and Voyager 2 probes.
Beyond this layered structure, the Sun is far more complex. Some other properties, which we describe shortly, are:
- The Sun vibrates. As a self gravitating compressible sphere, it vibrates. Pressure and density fluctuations mainly generated by the turbulent convection, are propagated through the Sun. Waves with frequencies and wavelengths close to those of the many normal modes of vibration of the Sun add up to a characteristic pattern of constructive
%?doppler shifts of p modes are about this order, rigth?
interference. This vibration, although of low amplitude with few 100 m/s in the photosphere, can be measured and decomposed into eigenmodes by means of Doppler shifts and observations of long duration. The propagation of the waves depends on the properties of the medium. It is possible then to infer these properties from the measured vibration patterns. Some waves propagate only close to the surface, but others can propagate through the entire Sun. These latter waves provide means to infer some structural properties, such as temperature, of the solar interior and test models of the Sun.
%franz i remove it : Nonetheless the velocitiy of this waves depends on the density, meaning that they carry less information of the inner parts.
\emph{Global Helioseismology} provides means to infer the global properties of the interior of the Sun studying the vibration pattern, while \emph{local helioseismology} can depict the surroundings of the local perturbations.
- The Sun rotates. The conservation of angular momentum of a slowly rotating cloud that will form a star result, upon contraction, a rapid rotation. It is commonly accepted that most of the Sun's angular momentum was removed during the first phases of the life of the Sun by braking via magnetic fields anchored in the surrounding interstellar medium and by a strong wind. The remaining angular momentum leads to today's solar rotation period. But being the Sun not a rigid body this rotation varies from layer to layer and with latitude. Gas at the equator rotates at the surface with a period of 27 days, faster than at the poles where the rotation period is approximately 32 days. Using helioseismology observations we know that this differential rotation continues inside the Sun, until a certain depth, from which on the inner part rotates like a rigid sphere with a period of that at middle latitudes on the surface. This region corresponds to the layer where the convection starts, at around $0.7$ solar radii, and is called the \emph{tachocline}. The differential rotation creates meridional flows of gas directed towards the poles near the surface and towards the equator near the bottom of the convection zone.
- The Sun shows (complex) magnetic activity. The Sun possesses a very weak overall magnetic dipole field. However, the solar surface can host very strong and tremendously complicated magnetic structures, which can be seen through their effects on the solar plasma, e.g. less efficient energy transport (that leads to dark sunspots). All matter in the Sun is in the form of plasma, due to the high temperature. The high mobility of charges that characterizes the plasma state, makes it highly conductive, causing magnetic field lines to be "frozen" into it. Provided that the gas pressure is much higher than the magnetic pressure, the magnetic field lines follow generally the dynamics of the plasma. The source of these localized strong magnetic fields is still to be understood. The dynamo theory addresses this problem suggesting that the weak dipolar magnetic field is amplified at the bottom of the convection zone by the stochastic mass motion and shear produced by the convection and the differential rotation.
- The Sun has cycles. The Sun suffers fluctuations in time. Changes occur in the total irradiance, in solar wind and in magnetic fields. They happen in approximately regular cycles, like the 11 years sunspot cycle, and aperiodically over extended times, like the Maunder Minimum (a period of 75 years in the XVII century when sunspots were rare, and which coincided with the coldest part of the \emph{Little Ice Age}). These fluctuations modulate the structure of the Sun's atmosphere, corona and solar wind, the total irradiance, occurrence of flares and coronal mass ejections and also indirectly the flux of incoming high-energy cosmic rays. None of these variations are fully understood and their effect on the Sun itself or Earth is still under debate. The generally accepted idea about the cyclic and more aperiodic fluctuations is that they are caused by variable magnetic fields. These are generated by dynamo mechanisms.
- The Sun evolves. The Sun is now in its main-sequence phase, where the main source of energy is the nuclear fusion of hydrogen to helium. After the initial phase of accretion of mass, a self gravitating star enters this phase, which lasts for most of its life. In the case of the Sun this phase will continue for approximately another five million years, after which the later evolution stages include a complex variation of the radius, with burning of helium as the source of energy in a later red giant phase. After this stage, the mass of the Sun is believed to be not large enough to undergo further fusion stages, and the Sun will slowly faint as a white dwarf star.
Readers can find further general information about the Sun in e.g. \cite{wikisun,Stix:2002lr} and many references therein.
\section{The chromosphere\label{intro:chromo}}
In our short description of the Sun's structure we stated that the atmosphere of the Sun comprises a layer above the photosphere in which the temperature begins to rise again until the transition region where an abrupt increase of temperature, from approximately 10\,000~K to 1 million K, occurs. This first layer above the photosphere is called \emph{chromosphere}. The name comes from the greek of ``color sphere'', as it can be seen as a ring of vivid red color around the Sun during total solar eclipses\footnote{The apparent size of the Sun on the sky happens to be very similar to the apparent size of the Moon, leading to annular or total solar eclipses, during which the red ring can be seen.}.
The boundaries of the chromospheric layer are very rugged, resembling more cloud structures than a spheric surface. Above quiet Sun regions the chromosphere can be about 2\,000 km thick, but some structures seen in typical chromospheric lines can reach to much higher altitudes, like filaments (that can reach heights of $350\,000$ km).
The solar chromosphere is a highly dynamic atmospheric layer. At most wavelengths in the optical range, it is transparent due to the fact that its density is low, much lower than in the photosphere below it. Nevertheless, in strong lines like H$\alpha$ (at 6563 \AA) or \ion{Ca}{II} K and H (at 3934 \AA\, and 3969 \AA, respectively) we have strong absorption (and re-emission) which allows direct studies about its peculiar characteristic, like bright plages around sunspots, dark filaments across the disk, as well as spicules and prominences above the limb. Indeed, recent works, e.g. \cite{2003A&A...402..361T}, suggest that many of these chromospheric features could all have the same physical properties but within different scenarios.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{../figures/mosaicn.png}
\caption{High resolution filtergram taken in the center of the H$\alpha$ spectral line, showing the chromosphere of the Sun with an image resolution of $\sim150$ km. The same field of view as image \ref{fig:photosphere}. The localized strong magnetic fields causing sunspots in the photosphere are seen now as fibrils around the sunspots. Given the low $\beta$ parameter, the plasma is forced to follow the magnetic lines, providing visible tracers and the variety of structures seen in the chromosphere. In the image we can see a carpet of spicules, plage region and a top view of a rising twisted magnetic flux tube above the active region. This image corresponds to the dataset ``sigmoid'' studied in Chapter \ref{chapter:hr}.}
\label{fig:chromosphere}
\end{center}
\end{figure}
The temporal evolution of the chromospheric structures is complex. The dynamics of a magnetised gas depends on the ratio of the gas pressure $P_\mathrm{gas}$ to the magnetic pressure $P_\mathrm{mag}$, i.e. the plasma $\beta$ parameter, $\beta$\,=\,$P_\mathrm{gas}/P_\mathrm{mag}$, with
$P_\mathrm{mag}$\,=\,$B^2/(8\pi)$ and $B$ the magnetic field strength \footnote{It is very common in astrophysics, specially in solar physics, to use magnetic field strength synonymously with magnetic flux density. The reason is that in most astrophysical plasmas B=H in Gaussian units. We follow this use in this thesis.}. From the low chromosphere into the extended corona, this plasma parameter decreases from values $\beta>1$, where the magnetic lines follow the motion of the plasma (as in the photosphere and solar interior) to a
low-beta regime, $\beta\ll1$, where the plasma motions are magnetically driven, and the plasma follows the magnetic field lines, creating visible tracers of the magnetism. These effects give rise to a new variety of energy transport and phenomena, like magnetic reconnection, filaments standing high above the chromosphere or erupting prominences.
%\begin{comment}
%\clearpage
\section{Aim and outline of this work\label{ref:out}}
Since the discovery of the chromosphere and since the hand-drawings of \citet{secchi1877} we have been able to observe this solar atmospheric layer in much detail. Many theoretical models have been proposed to understand its peculiar characteristics. But, only in the last recent years we have been able to address the problem with fine spectropolarimetry and high spatial resolution. We can study the fine details and resolve small structures, following their dynamics in time. Within these recent advances it has been possible both to test current theories and to observe new unexpected phenomena. This work thus aims at contributing to the understanding of the solar chromosphere.
This first Chapter provided a broad introduction to the context of this work. We have briefly presented some general properties of the Sun and the chromosphere. In the following pages, throughout Chapter 2, we summarize some theoretical concepts of radiative transfer and spectral line formation needed for this work. We also present general characteristics of the two spectral lines studied: H$\alpha$ and \ion{He}{i} 10830\, \AA. Chapter 3 presents in detail the observations. There we also summarize the characteristics of the used telescope and optical instruments, as well as the data reduction and post-processing methods applied to achieve spatial resolutions better than $0\farcs5$.
Next, in Chapter 4, we discuss results from data on the solar disc, dealing with the chromospheric dynamics and fast events observed in our data. We present the observations of magnetoacustic waves as well as other fast events. Chapter 5 is devoted to the spicules above the solar limb. The analysis of the spectroscopic intensity profiles from spicules in the infrared spectral range can be used to compare current theoretical models with observations. Further, we present high resolution images in H$\alpha$ of spicules. Finally, the concluding Chapter 6 of this thesis summarizes the main conclusions and gives an outlook for future work.
%\end{comment}
%%
\chapter{Spectral lines}
Most of the information from the extraterrestrial cosmos, also from the Sun, arrives as radiation from the sky. It comes encoded in the dependence of the intensity on direction, time and wavelength. Also, the polarization state of the light contains information. These characteristics of the light we observe from any object have their origin in the interaction of atoms and photons under the local properties (temperature, density, magnetic field, radiation field itself, \dots).
To extract this encoded information from the recorded intensities it is important to understand how the radiation is created and transported in the cosmic plasmas and released into the almost empty space.
This Chapter describes in the following sections the basis of radiative transfer and spectral line formation. We continue discussing the special properties of the spectral lines used in this work: the hydrogen Balmer-$\alpha$ line (named H$\alpha$ for short) at 6563~\AA, and the \mbox{\ion{He}{i} 10830\, \AA}\, multiplet. %Since the polarization of light carries also important information we introduce the Stokes parameters which describe the polarization state of the light.
\section{Radiative transfer and spectral line formation}
Light, consisting of photons, interacts with the gas (of the solar atmosphere, in our case) via absorption and emission. Let $I_{\lambda}(\vec{r},t,\vec{\Omega})$ be the specific intensity (irradiance) at the point $\vec{r}$ in the atmosphere, at time $t$, and into direction $\vec{\Omega}$, with $|\vec{\Omega}|=1$. We further denote by $\kappa_{\lambda}$ and $\epsilon_{\lambda}$ as the absorption and emission coefficients, respectively.
Along a distance $\mathrm{d}s$ in the direction $\vec{\Omega}$, the change of $I_{\lambda}$ is given by
\begin{equation}
\mathrm{d}\,I_{\lambda}= -\kappa_{\lambda}I_{\lambda} \mathrm{d}s + \epsilon_{\lambda} \mathrm{d}s\, ,
\end{equation}
or
\begin{equation}
\frac{\mathrm{d}\,I_{\lambda}}{\mathrm{d}s}= -\kappa_{\lambda}I_{\lambda}+ \epsilon_{\lambda}\, .
\label{radtran}
\end{equation}
We define also the optical thickness between some points $1$ and $2$ in the atmosphere by
\begin{eqnarray}
\mathrm{d}\,\tau_{\lambda}= -\kappa_{\lambda} \mathrm{d}s &;& \tau_{\lambda,{1}}-\tau_{\lambda,{2}}=-\int_{2}^{1}\kappa_{\lambda}\mathrm{d}s \, ,
\end{eqnarray}
and the source function $S_{\lambda}$ of the radiation field as
\begin{equation}
S_{\lambda}= \frac{ \epsilon_{\lambda}}{\kappa_{\lambda}}\, .
\end{equation}
In the solar atmosphere, absorption and emission are usually effected by transitions between atomic or molecular energy levels, i.e. by bound-bound, bound-free and free-free transitions. If collisions among atoms and with electrons occur much more often than the radiative processes, the atmospheric gas attains statistical thermal properties such as Maxwellian velocity distributions and the population and ionization ratios according to the Boltzamnn and Saha formulae. These properties define locally a temperature $T$. It can be shown (e.g. \citealt{Chandrasekhar:1960lr})
that in these cases, called \emph{Local Thermodynamic Equilibrium} (LTE), the source function is given by the Planck function or black body radiation
\begin{equation}
S_{\lambda}=B_{\lambda}=\frac{2hc^{2}}{\lambda^{5}}\frac{1}{e^{hc / \lambda k T}-1}\, .
\end{equation}
$S_{\lambda}$ varies much more slowly with wavelength than the absorption/emission coefficients across a spectral line. Thus, within a spectral line, $S_{\lambda}$ can be considered independent of $\lambda$.
Generally, LTE does not hold, especially in regions with low densities (thus with only few collisions relative to radiation processes) and near the outer boundary of the atmosphere from where the radiation can escape into space. The solar chromosphere is a typical atmospheric layer where non-LTE applies. In this case, the population densities of the atomic levels for a specific transition depend on the detailed processes and routes leading to the involved levels.
Equation \ref{radtran} has the following formal solution
\begin{equation}
I(\tau_{2})=I(\tau_{1})e^{-(\tau_{1}-\tau_{2})}+ \int_{\tau_{2}}^{\tau_{1}}S(\tau')e^{-(\tau'-\tau_{2})}\,d\tau' \, ,
\end{equation}
or, for the case when $\tau_{1} \rightarrow \infty$ (optically very thick atmosphere) and $\tau_{2}=0$ \,($I(\tau_{2}=0) \Rightarrow$ emergent intensity), then
\begin{equation}
I_{\lambda}(\tau_{\lambda}=0)=\int_{0}^{\infty}S_{\lambda}(\tau'_{\lambda})e^{-\tau'_{\lambda}}\,d\tau_{\lambda}' \, .
\end{equation}
A second order expansion of $S(\tau_{\lambda})$ leads to the Eddington-Barbier relation
\begin{equation}
I_{\lambda}(\tau_{\lambda}=0) \approx S_{\lambda}(\tau_{\lambda}=1)\,.
\end{equation}
This says that the observed intensity $I_{\lambda}$ at a wavelength $\lambda$ is approximately given by the source function at optical depth $\tau_{\lambda}=1$ at this same wavelength. In LTE, the intensity then follows the Planck function $B_{\lambda}(T(\tau_{\lambda}=1))$.
In spectral lines, the opacity is much increased over the continuum opacity. Since the temperature decreases with height in the solar photosphere the intensity in spectral lines is decreased, and we probe higher and cooler layers. This explains the formation of absorption lines in LTE.
In non-LTE, when collisional transitions between atomic levels occur seldom and near the outer atmospheric border, photons can escape and are thus lost for the build-up of a radiation field in the specific transition. Then, the upper level of the transition becomes underpopulated and the source function has decreased below the Planck function at the local temperature. It follows that, even for constant temperature atmospheres, a strong absorption line can be observed.
Outside the solar limb, in the visible spectral range, one observes spectral lines (and very weak continua) in emission. In spectral lines, high chromospheric structures are seen in front of a dark background.
\section{Hydrogen Balmer-$\alpha$ line (H$\alpha$)}
H$\alpha$ at $6563$ \AA\, is a strong absorption line in the solar spectrum for two reasons: 1) hydrogen is the most abundant element in the Sun, and in the Universe. 2) The Sun, as a G2 $\mathrm{V}$ star, has the appropriate effective temperature $T_{eff}\approx5\,800$ K to have the second level of hydrogen populated and thus to make absorptions in H$\alpha$ possible.
As all strong lines, H$\alpha$ possesses a so-called Doppler core and damping wings. The Doppler core of H$\alpha$ and of other Balmer lines is much broader than of other strong lines from metals (atomic species with $Z>2$). The reason is the large thermal velocity of hydrogen compared to that of metals, thus leading to large Doppler widths
\begin{equation}
\Delta\lambda_{D}=\frac{\lambda_{0}}{c}\sqrt{\frac{2\,\mathcal{R}\,T}{\mu}}\, ,
\label{dopwi}
\end{equation}
where $\lambda_{0}$ is the rest wavelength, $c$ the speed of light, $\mathcal{R}$
the universal gas constant, $T$ the temperature and $\mu$ the atomic weight (H has the minimum value among the chemical elements of $\mu = 1.008$). An eventual ``microturbulent'' broadening has been omitted in Eq. \ref{dopwi}.
Another property of the Balmer transitions between the according hydrogen levels is the following: Chromospheric lines such as the \ion{Ca}{ii} H and K and the \ion{Mg}{i} h and k lines are weakly coupled to the local temperature through collisional transitions, effected by electrons, between the involved energy levels. Thus, these lines still contain information about the temperature of the electrons, although only in a ``hidden'' manner.
However, for the Balmer lines of hydrogen and here especially for H$\alpha$, there exist also the routes for level populations through radiative ionization to the continuum and radiative recombination. These routes are taken much more often than the collisional transitions between the involved levels. The ionizing radiation fields, i.e. the Balmer and Paschen continua, originate in the lower to middle photosphere and are fairly constant, irrespective of the chromospheric dynamics. Only when many high-energy electrons, as during a flare, are injected into the chromosphere the H$\alpha$ line reacts to temperature and gets eventually into emission.
Nonetheless, the chromosphere observed in H$\alpha$ exhibits rich structuring, due to absorption by gas ejecta, due to Doppler shifts of the H$\alpha$ profile in fast gas flows along magnetic fields, and due to channeling of photons around absorbing features.
\section{\ion{He}{i} 10830 \AA\, multiplet \label{sec:limb:he}}
Helium is the second most abundant element in the Universe, also in the Sun. It was first discovered in the Sun in 1868 (from where it was named after the greek word of Sun).
At the typical chromospheric temperatures there is not enough energy to excite electrons to populate the upper levels from where these transitions occur. In coronal holes the helium lines are substantially weaker compared with the quiet Sun outside the limb. More information about recent advances in measuring chromospheric magnetic fields in the He I 10830 \AA\, line can be found in \cite{2007AdSpR..39.1734L}.
The energy levels that take part in the transitions of the \ion{He}{i} 10830 \AA\, multiplet are basically populated via an ionization-recombination process \citep{1994isp..book...35A}. The much hotter corona irradiates at high energies both outwards to space and inwards, to the chromosphere. The EUV coronal irradiation (CI) at
wavelengths lambda $\lambda<504$~\AA\ ionizes the neutral helium, and subsequent recombinations of singly ionized helium with free electrons lead to an overpopulation of the upper levels of the \ion{He}{i} 10830 multiplet.
Alternative theories suggest other mechanisms that may also contribute to the formation of the helium lines via the collisional excitation of the electrons in regions with higher temperature \citep[e.g.][]{1997ApJ...489..375A}.
\begin{wrapfigure}[18]{r}{0.5\textwidth}
\vspace{-1.1cm}
\begin{center}
\includegraphics[width=0.5\textwidth]{../figures/diagrams005.png}
\caption{Schematic Grotrian diagram for the \ion{He}{i} 10830~\AA\ multiplet emission lines.}
\label{fig:he:levels}
\end{center}
\end{wrapfigure}
The \ion{He}{i} 10830~\AA\ multiplet consists of the three transitions of the orthohelium (total spin of the electrons $S$=1) energy levels, from the upper term with angular momentum $L=1$ to the lower with $L=0$, in particular from $^3$P$_{2,1,0}$, which has three sublevels ($J=2,1,0$), to the lower metastable term $ ^3$S$_{1}$, which has one single level ($J=1$) (see Fig. \ref{fig:he:levels}). The two transitions from the J=2 and J=1 upper levels appear mutually blended, i.e. as merely one line, at typical chromospheric temperatures, and form the so-called red component, at 10830.3~\AA. The two red transitions are only 0.09~\AA\ apart. The blue component, at 10829.1~\AA, corresponds to the transition from the upper level with J=0 to the lower level with J=1.
The formation height of these lines is believed to be between 1\,500 and 2\,000 km, (e.g. \citealt{Centeno06}) although, as we already mentioned, the chromosphere is strongly rugged. The Land\'e factors of the lines are not zero, meaning that they are sensitive to external magnetic fields.
A more detailed description about the properties of the \ion{He}{i} 10830 multiplet, in particular related to the emission profiles
%and their polarization
observed in spicules above the limb is given in Chapter~\ref{ch:spicules}.
%tepper
\begin{comment}
\section{Stokes parameters}
Only work on this if I make use of Q,U,V right?
To fully describe the properties of a radiation field we need not only the intensity and its dependence with wavelength, but also the polarization status. Light, being a electromagnetic transversal wave, has its plane of vibration perpendicular to the direction of propagation. The polarization state is the measurement of any preferred axis of the direction of vibration of the electric vector field.
Let us consider a photon propagating along the $\vec{z}$ direction. We choose a set of $\{\vec{x},\vec{y}\}$, orthonormal vectors defining a plane perpendicular to $z$. The vibration of the electric field on this plane can be then described in the general case of an ellipse (see Fig. \ref{figstokesfig1}) as
\begin{eqnarray}
E_{x}(t)=\epsilon_{x}e^{-i\omega t}=A_{x}e^{i(\phi_{x}-\omega t)},\\
E_{y}(t)=\epsilon_{y}e^{-i\omega t}=A_{y}e^{i(\phi_{y}-\omega t)},
\label{stokes1}
\end{eqnarray}
where $\omega$ is the angular frequency and $\epsilon_{x},\epsilon_{y}$ are two complex numbers that can be written in terms of the real numbers $A_{x},A_{y},\phi_{x},\phi_{y}$. These values set thus the polarization status of the wave. On the general case where the vector describes an ellipse the direction of rotation of the vibration axis is defined as the sign of
\begin{equation}
\delta=\phi_{x} -\phi_{y}
\end{equation}
where $\delta>0$ means anti-clockwise (incoming radiation) and $\delta<0$ means clockwise rotation.
Each plane wave constituting a radiation field, i.e. each photons, is fully polarized as parametrized by the complex numbers $\epsilon_{x},\epsilon_{y}$. However, in the Nature we always find a mixture of many different polarization signals, creating a superposition of polarization states. To measure the total polarization status we can then make use of 4 parameters as defined by Sir George Stokes, that characterize in a unique way the properties of the resulting polarization ellipse:
\begin{eqnarray}
I=\langle \epsilon_{x}^{*}\epsilon_{x}\rangle+\langle \epsilon_{y}^{*}\epsilon_{y}\rangle \\
Q=\langle \epsilon_{x}^{*}\epsilon_{x}\rangle-\langle \epsilon_{y}^{*}\epsilon_{y}\rangle\\
U=\langle \epsilon_{x}^{*}\epsilon_{y}\rangle+\langle \epsilon_{y}^{*}\epsilon_{x}\rangle \\
V=i(\langle \epsilon_{x}^{*}\epsilon_{x}\rangle-\langle \epsilon_{y}^{*}\epsilon_{y}\rangle)
\label{stokes2}
\end{eqnarray}
where the symbols $\langle\rangle$ mean statitical mean over all the photons constituting the incident light.
Further, it is not possible to measure directly the amplitudes and phases of the electromagnetic vectors. We can instead measure the properties of the polarization ellipse by means of optical devices like polarizers or retarders. An adequate combination of measurements with these elements allows the direct measurement of the Stokes parameters. In Fig. ?? we represent the operation definition of the Stokes parameters in terms
Natural or non-polarized light occurs when the direction of vibration of electromagnetic vector changes randomly. Totally polarized light is the extreme case when this direction describes a certain trajectory in time, either fixed over a certain axis or rotating describing an ellipse on the plane perpendicular to the direction of propagation.
\begin{wrapfigure}[15]{r}{0.5\textwidth}
\vspace{-1.1cm}
\begin{center}
\includegraphics[width=0.5\textwidth]{../figures/diagrams005.png}
\caption{blablabla}
\label{figstokesfig1}
\end{center}
\end{wrapfigure}
For a more detailed ontro of th Soktes parameters we refeer to \cite[e.g.][]{Chandrasekhar:1960lr,Born:1999lr}.
In absence of external magnetic fields or isotropic radiation, the selection rules populate equally all sub-levels, so the emitted radiation is unpolarized. However, an external magnetic field can split the sublevels, and therefore the energy transitions, no with different polarizations (Zeeman effect). Also the mere anisotropic radiation pumping can produce population imbalances and therefore polarization signals (Hanle effect), as we will show in Sec. \ref{hanle}.
\end{comment}
\chapter{Observations}
For the present work we used data from two different instruments, both mounted on the same telescope, the \emph{ Vacuum Tower Telescope} (VTT, Sec. \ref{sec:telescope}) in Tenerife. One of the instruments, the \emph{G\"ottingen Fabry-Perot Interferometer} (G-FPI, Sec. \ref{obs:fpi}) is able to achieve very high spatial resolution while the other, the \emph{Tenerife Infrared Polarimeter} (TIP, Sec. \ref{obs:tip}), is able to obtain full Stokes spectropolarimetric data with very high spectral resolution. Both instruments, in combination with the \emph{Kiepenheuer Adaptive Optics System} (KAOS, Sec. \ref{obs:kaos}), provided the data for this work.
In this Chapter we will describe the telescope, the instrumentation, the observations, and the reduction techniques. The latter are aimed at removing as many instrumental effects as possible.
\section{Angular resolution and \emph{Seeing}\label{seeing}}
When using any kind of an optical imaging system, the angular resolution in the focal plane is limited by diffraction at the aperture of the instrument. For circular apertures the image of a point source (the PSF) is an Airy function with a certain Full Width at Half Maximum (FWHM). Two point sources closer than the FWHM of a certain instrumental PSF are difficult to distinguish. If one considers diffraction of a telescope with a circular aperture of diameter $D$, the angular resolution limit is, in the usual Rayleigh definition,
\begin{equation}
\alpha_{min}=1.22 \, \frac{ \lambda}{D} \, .
\label{ec:res}
\end{equation}
The factor 1.22 is approximately the first zero divided by $\pi$ of the Bessel function involved in the Airy function. In the focal plane of such a telescope with a focal length $f$ the spatial resolution is therefore $d=1,22 \, \lambda\, f/D$. For good sampling this should correspond to, or even be larger than, the resolution element of the detector (2 pixels). In the case of the VTT, with a main mirror of $D=70$ cm, the diffraction limited resolution is $0\farcs24$ at $6563$ \AA\,(H$\alpha$) and $0\farcs39$ at $10830$ \AA\, (\ion{He}{i} triplet). In solar observation, it is common to use as the diffraction limit simply $\alpha_{min}=\lambda/D$. At this angular distance the modulation transfer function (MTF) has become zero.
Unfortunately all imaging systems on the ground are subject to aberrations that degrade the image quality, resulting in a much lower spatial resolution than the diffraction limit. The light we observe from the Sun travels unperturbed along approximately 150 million km, but during the last few microseconds before detection it becomes distorted due to its interaction with the Earth's atmosphere and our optical instrument.
%\pagebreak
The refraction index of the air is very close to 1 at optical wavelengths, but depends on the local pressure and temperature. Their fluctuations in space and time produce aberrations of the wavefronts from the object to be observed\footnote{The local values of the temperature and pressure depend on the complicated turbulent dynamics of the atmosphere. This includes friction and heating of the Earth's irregular surfaces, condensations and formation of clouds, shears produced by strong winds, \dots For more information we refer to e.g. \citealt{2002RvMP...74..551S}.}. Since the time scale of the variation of the aberrations of $\approx 10$ ms is usually smaller than the integration time, it also produces smoothing of the image details. Thus, the information at small scales is lost.
Further, the turbulent state of the air masses through which the light is passing varies on small angular scales. This produces an anisoplanatism of the wavefronts arriving at the telescope, with angular sizes of the isoplanatic patches not larger than approximately $ 10\arcsec$.
Beside the atmospheric factors, the final quality of the image is influenced by local factors like the aerodynamical shape of the telescope building or convection around the building and the dome.
Finally, the internal \emph{seeing} of the telescope plays an important role for the image quality. Convection along the light path in the telescope triggered by heated optical surfaces can be avoided by allowing air flowing freely through the structure or, quite the contrary, by evacuating the telescope.
In solar physics we usually measure the average image quality of the observations estimating the diameter of a telescope that would produce, from a point source, an image with the same diffraction-limited FWHM as the atmospheric turbulence or internal \emph{seeing} would allow even with a much larger telescope aperture. This is called the Fried parameter ($r_{0}$). Typically, upper limits for the ``Observatorio del Teide'' are $r_{0} \approx 15$\, cm during night-time and $r_{0} \approx 7$\, cm during day.
Besides these structural requirements for best \emph{seeing} conditions, there are nowadays methods for correcting the images for seeing distortions to obtain near diffraction limited resolution. In this thesis we have used various methods: We correct partially the aberrations in real time using adaptive optics (Sec. \ref{obs:kaos}) which can increase the $r_{0}$ around the center of the field of view up to $r_{0}\sim 25$\,cm and we also apply post-processing methods of image reconstruction (Sec. \ref{datared}) to approach the upper limit of $r_{0} \lesssim 70$ cm.
\newpage
\section{Telescope\label{sec:telescope}}
The \emph{Vacuum Tower Telescope} \citep[VTT,][ Fig. \ref{fig:foto:vtt}]{1985spit.conf.1191S} is located at the Spanish ``Observatorio del Teide'' (2400 m above sea level, 16\fdg 30' W, 28\fdg 18' N) in Tenerife, Canary Islands. It is operated by the Kiepenheuer-Institut f\"ur Sonnenphysik, Freiburg, with contributions from the Institut f\"ur Astrophysik in G\"ottingen, the Max-Planck-Institut for Sonnensystemforschung, Katlenburg-Lindau, and the Astrophysikalisches Institut Potsdam.
\begin{wrapfigure}[19]{r}{0.4\textwidth}
\vspace{-0.4cm}
\begin{center}
\includegraphics[height=7cm]{../figures/VTT.jpg}
\caption{Building which houses the solar \emph{Vacuum Tower Telescope}.}
\label{fig:foto:vtt}
\end{center}
\end{wrapfigure}
The VTT optical setup is depicted in Fig. \ref{fig:vtt:optical}. At the top platform of the building, a coelostat achieves to follow the path of the Sun on the sky, by means of two flat mirrors of very high optical quality. The primary coelostat mirror rotates clockwise (seen pole-on) about an axis which is contained in the mirror surface and is parallel to the Earth's rotation axis. It reflects the sunlight towards the secondary mirror. The latter redirects the beam towards the fixed telescope in the tower. The telescope is an off-axis system. It consist of a slightly aspherical main mirror of 70 cm diameter and a focal length of 46 m, and of a folding flat mirror. The free aperture of the circular entrance pupil with D=70 cm gives the telescopic diffraction limit for the angular resolution of $\alpha_{min}=\lambda/D \approx0\farcs16$ for $\lambda$ in the visible spectral range.
To avoid turbulent air flows inside the telescope caused by heated surfaces, the telescope is mounted in a tank that is evacuated to 1 mbar. The vacuum tank has high quality transparent entrance and exit windows located below the coelostat and close to the primary focus, respectively.
Shortly after the entrance window, a small part of the sunlight is reflected out to a second imaging device. This uses a quadrant cell to track the image of the solar disc and to correct slow image motions, e.g. due to a non-perfect hour drive of the coelostat. Telescope pointing to a target inside and near the solar disc is achieved by moving this tracking device as a whole in the image plane. The imbalanced illumination of the quadrant cell is transformed to a tip-tilt motion of the secondary coelostat mirror.
After the main vacuum tank, the adaptive optics (Sec.\ref{obs:kaos}) device is located. This optical system is able to correct in real time the low order aberrations of the incoming wavefronts of the light beam caused by the turbulence in the Earth's atmosphere. After the adaptive optics system, which can optionally be moved in or out of the path, the light path continues to the vertical slit spectrograph or to a folding mirror that can be used to direct the light to different other available science instruments.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.5\textwidth]{../figures/diagrams001.png}
\caption{Optical setup of the VTT. The coelostat (mirrors \emph{m1,m2}) follows the path of the Sun on the sky and directs the light to the entrance window of the vacuum tank (blue shaded). Mirror \emph{m3} takes out a small amount of the light and feeds the guiding telescope mounted outside the vacuum tank. The collimating mirror \emph{m5} produces, together with the flat mirror \emph{m6}, the solar image in the primary focal plane behind the exit window of the vacuum tank. There, a flat mirror can be mounted under $45^{\circ}$ to the vertical (not shown) to feed post-focus instruments in optical laboratories. The adaptive optics system is located below the exit window, and it is used optionally. }
\label{fig:vtt:optical}
\end{center}
\end{figure}
\subsection{Kiepenheuer Adaptive Optics System\label{obs:kaos}}
As mentioned in the beginning of this Chapter (Section \ref{seeing}) the atmosphere of the Earth degrades the quality of the images during observations. KAOS \citep[Kiepenheuer Adaptive Optics System, ][]{2003SPIE.4853..187V,2007msfa.conf..107B} is a realtime correction device that calculates and corrects the instantaneous aberrations of the wavefront using special deformable mirrors.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{../figures/diagrams002.jpg}
\caption{Scheme of typical AO. Inside the closed loop, a fraction of the incoming light is directed to the KAOS camera (semitransparent mirror \emph{m1}), where a lenslet array (\emph{ll}) produces many subfield images with light from different parts of the pupil. The calculated instantaneous aberration is compensated using the two (tip\&tilt and deformable) mirrors, every 0.4 ms.}
\label{fig:kaos:optical}
\end{center}
\end{figure}
The optical scheme of a typical adaptive optics (AO) system is shown in Fig. \ref{fig:kaos:optical}. By means of a dichroic semitransparent beam splitter, part of the light entering the system is directed to the wavefront sensor. The latter, a Shack-Hartmann sensor, consists of a lenslet array positioned in an image of the entrance pupil and a fast CCD detector. Each lenslet, cutting out a subaperture of the pupil image, produces an image of a small area on the Sun on a subarea of the CCD. Using a good, i.e. as sharp as possible, subimage of the present scenery on the Sun and with a correlation algorithm, it is possible to compute the displacement of each subimage and to estimate from this the aberrations of the wavefront.
Every aberration can be expressed by a sum of adequate polynomials (for example Zernike polynomials) with appropriate coefficients. Each polynomial represents a specific wavefront aberration, e.g. tilt, defocus, astigmatismus \dots The AO is able to correct the low orders of the aberration, that is those with the largest scales. For this purpose it has two active optical surfaces (both of them in the main lightbeam, so the correction is done in a closed loop). In the case of KAOS the first element is the tip-tilt mirror that is able to displace the whole image in two perpendicular directions, thus tracking on the reference image. The second optical element is a bymorphous deformable mirror with 35 actuators. With appropriate voltages, the surface of this mirror obtains a shape that corrects the aberrations of the incoming wavefront up to the $27^{th}$ Zernike polynomial. This correction is done in a fast closed loop at 2100 Hz. The bandwidth of KAOS is 100 Hz. It thus operates at timescales comparable to that of the variation of the turbulence in the atmosphere.
As already mentioned, the aberration of the wavefront is not constant, i.e. not isoplanatic across the whole field of view (FoV). The wavefront camera has a restricted FoV of $12\arcsec \times 12\arcsec$ where the assumption of isoplanatism is approximately valid. The center of this subfield of AO correction is called \emph{lockpoint}. The restricted area of isoplanatism is one of the main limitations of current AO systems. The corrections are calculated for the lockpoint feature we are tracking on and applied to the whole FoV of the telescope. Therefore the correction becomes increasingly inaccurate with increasing distance from the lockpoint. The quality of the image is degraded outwards from the center of the FoV, where the \emph{lockpoint} is usually located. Fortunately this can be taken into account using {\em post factum} image reconstruction like speckle interferometry and blind deconvolution.
In night-time astronomy, AO systems lock on a star image, so the displacements of the subfields imaged by the lenslet are easily calculated. In solar observations, the image used by the AO comes always from an extended source, making the calculations of the displacements much more demanding. In solar AOs, a reference image is taken and updated regularly during operation, and correlations between this image and the subfield images are used. For well defined maxima of the correlation functions we need features with sufficient contrast inside the FoV to lock on with the algorithm, e.g. a pore or the granulation pattern. Moreover, the wavefront sensor can only work with a high light level, e.g. integrated over some wavelength. So it is not possible to lock for example on features within the H$\alpha$ line with low intensity. Also, as we will explain in Sec. \ref{obs:tip}, near or off-limb observations are difficult as the AO algorithm is not able to track on that kind of references, as the one-dimensional limb image.
\section{High spatial resolution}
For our study of the dynamics of chromospheric structures, we are interested in observations with the highest possible spatial resolution\footnote{It has become a widespread custom in solar observations to use ``spatial resolution'' synonymously with ``angular resolution''.}, with the highest achievable temporal cadence, and with as much spectral information as possible. For that purpose we used the ``G\"ottingen'' Fabry-Perot Interferometer (G-FPI). Here, the designation FPI stands as \emph{pars pro toto}, for the whole post-focus instrument, a two-dimensional spectrometer based on wavelength scanning Fabry-Perot etalons. It was developed at the Universit\"ats-Sternwarte G\"ottingen \citep{1992A&A...257..817B,1993PhDT.......243B,1995A&AS..112..371B}. Subsequently, it had undergone several upgrades \citep{2001A&A...365..588K, 2006A&A...451.1151P,Gonzalez:2007fk}. For the present work, the G-FPI with the high-efficiency performance described by \cite{2006A&A...451.1151P} was employed.
Basically, this instrument was able, at the time the data for this study were taken, to produce an image from a selected wavelength range with a narrow passband of 45 m\AA\,
%franz no era 55?
FWHM at 6563 \AA\, (H$\alpha$). A recent upgrade has reduced the FWHM. The spectrometer also can be tuned to almost any desired wavelength, being able to scan a spectral line, producing 2D filtergrams (images) at, e.g., 20 spectral position along a line. If we scan iteratively one spectral line we obtain a time sequence of very high spatial resolution, at several spectral positions and with a cadence which would be the time required to scan the full line, which is typically in the order of 20 seconds for our data.
The main limitation of this kind of observational procedure is that the images corresponding to a single scan are not obtained simultaneously, as they are taken consecutively. This is of special importance when we compare the images in the two wings of a spectral line, as the small-scale solar structure under study may have changed during the time needed to scan between these positions. This should be taken into account when studying features whose typical timescale of variation is comparable to the scanning time. In Sec. \ref{obser}, we will see that this limitation can partly be compensated when we have a long temporal series.
\subsection{Instrument\label{obs:fpi}}
The G\"ottingen Fabry-Perot Interferometer \citep{1995A&AS..112..371B,volkmer95,2001A&A...365..588K,2006A&A...451.1151P} is a speckle-ready two-dimensional (2D) spectrometer. It is able to scan a spectral line producing a set of speckle images at several spectral position with a narrow spectral FWHM, while taking simultaneous broadband images, needed for the {\em post factum} image reconstruction.
\subsubsection*{Fabry-Perot interferometer (FPI)}
A Fabry-Perot interferometer, or etalon, is an interference filter possessing two plane-parallel high-reflectance layers of high quality ($ \sim\lambda/100$). Light entering the filter is many times reflected between the plane-parallel reflecting surfaces. These reflections will produce destructive interference for transmitted light at all wavelengths but the ones for which two times the spacing $d$ of the plates is very close to a multiple of the wavelength. This effect gives rise to a final Airy intensity function \citep{Born:1999lr}:
\begin{equation}
I=I_{max}\frac{1}{1+\frac{4R}{(1-R)^{2}}\sin^{2}\frac{\delta}{2}} \, ,
\end{equation}
where the maximum intensity $I_{max}=\frac{T^{2}}{(1-R)^{2}}$ , $T$ is the transmittance, $R$ is the reflectance ($R=1-T$ if absorption is negligible), and the dependence on wavelength $\lambda$, angle of incidence $\Theta$, and refractive index $n$ of the material between the surfaces is
\begin{equation}
\label{eq:fpid}
\delta=\frac{4\pi}{\lambda}nd\,\cos\Theta \, .
\end{equation}
\begin{figure}[t]
\centering
\subfloat{\vspace{-0.2cm}
\includegraphics[width=0.5\textwidth]{../figures/scan-im.jpg}}%
\quad%
\subfloat{
\includegraphics[width=0.43\textwidth]{../figures/fts_fpiha.pdf}}
\caption{Example of the narrow-band scanning with the G-FPI. {\bf Left}: One narrow-band frame from a two-dimensional spectrometric scan through the hydrogen Balmer-$\alpha$ line (H$\alpha$).{ \bf Right}: H$\alpha$ line; \emph{solid black} from the Fourier Transform Spectrometer (FTS) atlas (Brault \& Neckel, quoted by \citealt{Neckel:1999lr}); \emph{blue}: FTS profile convolved with the Airy transmission function of the FPIs; \emph{dashed} average $H\alpha$ profile observed with the spectrometer at 21 wavelength position (\emph{rhombi}) with steps of 100 m\AA. The \emph{red} line is the Airy transmission function, positioned at the wavelength in which the image in the left panel was taken, and re-normalized to fit on the plot..}
\label{fpi:scan}
\end{figure}
The narrow transmittance of the filter can be tuned to any desired wavelength by changing the spacing $d$ (or the refractive index $n$, for pressure controlled FPIs).
One single FPI produces a channel spectrum according to the interference condition, i.e. for normal incidence ($\Theta = 0^{\circ}$) and assuming $n$=1,
\begin{equation}
m\lambda = 2 d
\end{equation}
with $m$ being the order. From here, the distance to the next transmission peak, or \emph{free spectral range (FSR)}, follows as
\begin{equation}
\emph{FSR}=\frac{\lambda^{2}}{2d} \, .
\end{equation}
To suppress all but the desired transmission, the G-FPI has a second Fabry-Perot etalon with different spacing, i.e. different \emph{FSR}. Both Fabry-Perot etalons need to be synchronized when scanning in order to keep the desired central transmittance peaks coinciding. The combination of two FPI with different \emph{FSR} removes effectively the undesired transmission peaks from other orders. An additional interference filter ($FWHM \approx 8\, \AA %franz check
$) is used to reduce the incoming spectral range to the spectral line under observation. The combination of these three elements produces a single narrow central peak, as depicted in Fig. \ref{fig:gfpi:transimance}.
The FP etalons are mounted close to an image of the telescope's entrance pupil in the collimated, i.e. parallel, beam. On the one hand, this avoids the ``orange peel'' pattern in the images, which one obtains with the telecentric mounting near the focus and which arises from tiny imperfections of the etalon surfaces. On the other hand, in the collimated mounting one has to deal with the fact that the wavelength position of the maximum transmission depends on the position in the FoV. This can be seen from Eq. \ref{eq:fpid} where the angle of incidence $\Theta$ changes with position in the FoV.
For the {\em post factum} image reconstruction (Sec. \ref{datared}) we have to acquire simultaneously short-exposure images from the narrow-band FPI spectrometer and broadband images. The latter are taken through a broadband interference filter ($FWHM \approx 50\, \AA $) at wavelength close to the one observed with the spectrometer. Two CCD detectors, one for each channel, with high sensitivity and high frame rates were used which allow a high cadence of short exposures. All processes (simultaneous exposures, synchronous FPI scanning and observation parameters) are controled by a central computer. The imaging on the two CCDs is aligned with special mountings and adjusted to have the same image scale on the two detectors.
The optical setup is shown schematically in Fig. \ref{fig:gfpi:optical}. From the focal plane following KAOS the image from the region of interest on the Sun is transferred via a $1:1$ re-imaging system into the optical laboratory housing the FPI spectrometer. In front of the focus at the spectrometer entrance, a beam splitter directs 5\% of the light into the broadband channel. The latter contains a focusing lens, the broadband interference filter (IF1), a filter blocking the infrared light (KG1, from \emph{Kaltglas} = ``cold glass'', notation by Schott AG), a neutral density filter to reduce the broadband light level, and a detector CCD1.
Most of the light (95 \%), enters the narrow-band channel of the spectrometer through a field stop at the entrance focus. After the field stop follow: an infrared blocking filter (KG2), the narrow interference filter (IF2), a collimating lens giving parallel light, the two Fabry-Perot etalons (FPI-B and FPI-N), a camera lens focusing the light on the detector CCD2. Figure \ref{fpi:scan} gives an example of the type of observation one can obtain with this narrow-band spectrometer.
The instrument has additional devices for calibration and adjustment: a feed of laser light, facilities to measure with a photomultiplier and to aid identifying the spectral line to be observed, and a feed of continuum light for various purposes, e.g. co-aligning the transmission maxima of the etalons or measuring the transmission curve of the pre-filter IF2.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{../figures/airy_filtro.pdf}
\caption{Transmission functions for the narrow-band channel of the G-FPI with the H$\alpha$ setup. The periodic Airy function of the narrow-band FPI (dashed line) coincides in the central wavelength with that of the broadband FPI (strong dashed green line). The global transmission of both FPIs has one single strong and narrow peak at the central wavelength (purple strong line). An additional interference filter (red line) is mounted to restrict the light to the scanned spectral line.}
\label{fig:gfpi:transimance}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[height=7cm]{../figures/diagrams003.png}
\caption{Schema of the ``Gottingen'' Fabry-Perot interferometer optical setup. After KAOS, the light is transferred from the telescope's primary focus to the spectrometer. A beam splitter BS directs 5\% of the light into the broadband channel consisting of a focusing lens L1, a broadband interference filter IF1 ($FWHM \approx 50\,\AA$), an infrared blocking filter KG1 (``Kaltglas''), a neutral density filter ND, and the CCD1 detector. 95\% of the light enter the spectrometer through a field stop at the entrance focus. Then follow: infrared blocking filter KG2, interference (pre-) filter IFII ($FWHM \approx 6 \AA\dots10\AA$, depending on the spectral line and wavelength range), collimating lens L2, the two FPI etalons FPI B and FPI N ($FWHM \approx 45 m\AA$ at H$\alpha$), the focusing camera lens L3 and the CCD2 detector. CCD1 and CCD2 take short-exposure (3-20 ms) images strictly simultaneously.}
\label{fig:gfpi:optical}
\end{center}
\end{figure}
\subsection{Observations\label{oberv}}
For the study of the chromospheric dynamics on the basis of high resolution observations we have used three data sets. Table \ref{table:obs:HRb} lists the details for each data set:
\begin{itemize}
\item Dataset \textit{mosaic} focuses on the study of a large active solar region, where we find fast moving dark clouds, as we will discuss in Sec. \ref{hr:darkclouds}. These data were obtained before the instrument upgrading in 2005 \citep{2006A&A...451.1151P} with the old cameras. The exposure time was six times longer than with the new CCDs and the FoV of a single frame is one fourth of that of the new version of the G-FPI. The observers of these data were M\'onica S\'anchez Cuberes, Klaus Puschmann and Franz Kneer.
\item Dataset \textit{sigmoid} uses the improvements of the instrument from 2005 and was obtained during excellent seeing conditions from a very active region. During the time span of our observations at least one flare was recorded from this region in our FoV. Our focus with these data is the study of fast events and magnetoacustic waves (Sec. \ref{waves1}) with the original intention to detect Alfv\'en waves. Examples of these data were also used to compare the results from different methods of \emph{post factum} image reconstruction, as we will show in Sec. \ref{sec:comp}.
\item With dataset \textit{limb} and in Sec. \ref{sec:comp} we apply blind deconvolution methods for image reconstruction (see Sec. \ref{momfbd}). The observations were taken with the G-FPI, renewed in 2005, to study with very high spatial resolution the evolution of spicules as seen in the H$\alpha$ line.
\end{itemize}
\begin{table}[b]
\begin{center}\begin{tabular}{|r|c|c|c|}\hline
\textbf{ Data set name} & \textit{``mosaic''} & \textit{``sigmoid''} & \textit{``limb''} \\\hline\hline
Date & May,31$^{st} $,2004 & April,26$^{th} $,2006 & May,4$^{th}$, 2005 \\\hline
Object & AR0621 & AR10875 & limb \\\hline
Heliocentric angle & $\mu=0.68$ &$\mu=0.59$ & $\mu=0$ \\\hline
Scans \# & 5 & 157 & 5 \\\hline
Cadence & 45 s & $\sim22$ s (see Sec. \ref{obser}) & $\sim19$ s \\\hline
Time span & 4 min & 55 min & 2 min \\\hline
Line positions \# & 18 & 21 & 22 \\\hline
FWHM & \multicolumn{3}{|c|}{50 \AA\, broadband / 45 m\AA\, narrow-band} \\\hline
Broadband filter & \multicolumn{3}{|c|}{6300 \AA} \\\hline
Stepwidth & 125 m\AA & 100\,m\AA & 93 m\AA \\\hline
Exposure time & 30 ms & \multicolumn{2}{|c|}{5 ms} \\\hline
Seeing condition & good & $r_{0} \approx 32$ cm & $r_{0} \approx 20$ cm \\\hline
KAOS support & \multicolumn{3}{|c|}{yes} \\\hline
Image reconstruction& speckle & AO ready speckle & MFMOBD \\\hline
Field of view & 33\arcsec $\times$ 23\arcsec (total 103\arcsec $\times$94\arcsec ) & \multicolumn{2}{|c|}{77\arcsec $\times$ 58\arcsec} \\\hline
\end{tabular} \caption{Characteristics of the data sets taken with the G-FPI used in this work.\label{table:obs:HRb}}
\end{center}
\end{table}
\subsection{Data reduction\label{datared}}
After the recording of the data, several processing steps have to be carried out in order to minimize the instrumental effects. These are mainly to take into account the differential sensitivity of the CCDs from one pixel to another or the fixed imperfections on the optical surfaces positioned close to one of the focal planes. This concerns for example dust on the beam splitter, on the infrared blocking filters and interference filters and the CCDs. In this step we also remove an imposed bias signal applied electronically to every frame. This is the usual treatment of any CCD data.
For this purpose we take flat fields, dark, continuum and target images (see Fig. \ref {fig:obs:red2}).
\begin{figure}[t]
\centering
\subfloat[Broad band raw frame]{
\includegraphics[width=0.45\textwidth]{../figures/raw.jpg}}%
\quad%
\subfloat[Flat field frame ]{
\includegraphics[width=0.45\textwidth]{../figures/flat.jpg}}
\\
\quad%
\subfloat[Dark frame ]{
\includegraphics[width=0.45\textwidth]{../figures/dark.jpg}}
\quad%
\subfloat[Reduced frame]{
\includegraphics[width=0.45\textwidth]{../figures/redu.jpg}}
\caption{Example of the standard data reduction process. Every frame taken with the CCD (a) includes instrumental artifacts like shadows from dust particles on the CCD chips or the filters near the focus (Fig. b) and the intrinsic differential response of each pixel (c). Subtracting the dark frame and dividing by the flat response provides a clean frame (d).}
\label{fig:obs:red2}
\end{figure}
\emph{Target}. A target grid is located in front of the instrument, in the primary focal plane. Target frames therefore display in both channels a grid of lines that are used to focus and align the cameras in both channels. This is crucial for the image reconstruction.
\emph{Continuum} data are taken with the same scanning parameters as with sunlight but using a continuum source, so we can test the transmission of the scanning narrow-band channel.
\emph{Dark} frames are taken with the same integration time but blocking the incident light. These frames have information of the differential and total response of the CCD array without light, in order to remove this effect from the scientific data.
\emph{Flat fields} are frames with the same scanning parameters and with sunlight, but without solar structures. In this way we can see the imperfections and dust on the optical surfaces fixed on every frame taken with the instrument, and remove them dividing our science data by these flat frames. To avoid signatures from solar structures in the flat frames, the telescope pointing is driven to make a random path around the center of the solar disc far from active regions.
Thus, to reduce the instrumental effects we use the following formula, for each channel and for each spectral position independently:
\begin{equation}
reduced frame=\frac{raw\,frame - mean\,dark}{mean\,flatfield - mean\,dark} \, .
\end{equation}
Our instruments produce data sets that can be subject to \emph{post factum} image reconstruction. We have applied speckle and blind deconvolution methods to minimize the wavefront aberrations and to achieve spatial resolution close to the diffraction limit imposed by the aperture of the telescope.
The aberrations are changing in time and space. In a long exposure image, the temporal dependence will produce the summation of different aberrations, blurring the small details of the image. Therefore, for post-processing, all image reconstruction methods need input \emph{speckle} frames with integration times shorter than the typical timescale of the atmospheric turbulence. With this condition fulfilled, the images appear distorted and speckled but not blurred, and still contain the information on small-scale structures. Another common characteristic of speckle methods is the way to address the field dependence of the aberrations. In a wide FoV each part of the frame is affected by different turbulences. That is, inside the atmospheric column affecting the image, there are spatial changes of the wavefront aberration. Therefore, the FoV is divided into a set of overlapping subfields smaller than the typical angular scale of change of the aberrations (5\arcsec -- 8\arcsec), the isoplanatic patch.
Speckle interferometry denotes the interference of parts of a wavefront from different sub-apertures of a telescope. This results in a speckled image of a point source, e.g. of a star. The effect is used for ``speckle interferometric'' techniques of postproccesing. They are able to remove the atmospheric aberrations of the wavefronts that degrade the quality of the images. In the following Sections we introduce the basic background of the methods used and provide some examples and further reference.
\subsubsection{Speckle interferometry of the broadband images\label{SIb}}
This method is based on a statistical approach to deduce the influence of the atmosphere. It was developed following the ideas of \cite{1965JOSA...55.1427F,1970A&A.....6...85L,1973JOSA...63..971K,1977OptCo..21...55W,von-der-Luehe:1984fk} . The code used for our data was developed at the Universit\"ats-Sternwarte G\"ottingen \citep{1996A&AS..120..195D} . The \emph{sigmoid} dataset uses the latest improvements to take into account the field dependence of the correction from the AO systems \citep{2006A&A...454.1011P}.
In what follows we present a brief overview of the method:
The observed image (\emph{i}) is the convolution ($\star$) of the true object (\emph{o}) with the \emph{Point Spread function ($PSF$)}. The $PSF$ is the intensity distribution in the image plane from a point source with intensity normalized to one, i.e.
\begin{equation}
\int\int PSF (x,y) dx dy = 1 \, ,
\end{equation}
where the integration is carried out in the image plane. The $PSF$ depends on space, time and wavelength. Its Fourier transform ($\mathscr{F}$) is the \emph{OTF, Optical Transfer Function}
\begin{equation}
\mathscr{F} ( i ) = \mathscr{F} (o \star PSF ) \hspace{0.5cm} \rightarrow \hspace{0.5cm} I=O \cdot OTF\, .
\label{ec:obs:obser}
\end{equation}
A normal long exposure image would be just the summation of N speckle images:
\begin{equation}
\sum^{N}_{i=1} I_{i} = O \cdot \sum^{N}_{i=1} OTF_{i} \, .
\label{ec:obs:long}
\end{equation}
The $OTF_{i}$ are continuously changing in time, which leads to a loss of information. The temporal phase change of the $OTF_{i}$ will, upon this summation, reduce strongly or even cancel the complex amplitudes at high wavenumbers. \cite{1970A&A.....6...85L} proposed to use the square modulus, to avoid cancellations:
\begin{equation}
\frac{1}{N}\sum^{N}_{i=1} |I_{i}|^2 = |O|^{2} \cdot \frac{1}{N} \sum^{N}_{i=1} |OTF_{i}|^2 = |O|^2 \cdot STF \, .
\label{ec:obs:stf}
\end{equation}
\noindent
Yet this procedure also removes the phase information on $o$. Thus, the phases have to be retrieved afterwards. \emph{STF} is the \emph{Speckle Transfer Function}, it contains the information on the wavefront aberrations during N speckle images. To deduce this STF is therefore one of the aims of the speckle method. On the Sun, point sources do not exist. It is thus not a trivial task to determine the $STF$. There are, however, models of $STF$ for extended sources from the notion that they depend only on the seeing conditions, through the \emph{Fried} parameter $r_{0}$ \citep{1973JOSA...63..971K}. This parameter can be calculated \emph{statistically} using the spectral ratio method \citep{von-der-Luehe:1984fk}. As this is a statistical approach, a minimum number of speckle frames must be used, more than 100.
To recover the phases of the original object the code uses the speckle masking method \citep{1977OptCo..21...55W,1983OptL....8..389W}. It recursively recovers the phases from low to high wavenumbers.
Finally a noise filter is applied, zeroing all the amplitudes at wavenumbers higher than a certain value, which depends on the quality of the data.
\begin{figure}[t]
\centering
\subfloat[Average of 330 speckle images (total exposure time $\sim1,6$ s). ]{
\includegraphics[width=0.47\textwidth]{../figures/broad-int.jpg}}
\quad%
\subfloat[Single speckle frame, 5 ms exposure time.]{
\includegraphics[width=0.47\textwidth]{../figures/broad-speckle.jpg}}%
\\
\quad%
\subfloat[Reconstructed broadband image, using 330 speckle frames. ]{
\includegraphics[width=\textwidth]{../figures/broad-redu.jpg}}
\caption{Example of improvement of broadband images with the speckle reconstruction. The size of the image is $\sim$ 34\arcsec $ \times $ 19\arcsec. The achieved spatial resolution is close to the diffraction limit, $ 0\farcs22$, with the diffraction limit $\alpha_{min}=\lambda/D \, \hat{=}\, 0\farcs19$ at $\lambda=6563$ \AA\,(H$\alpha$) and telescope aperture $D=70$~cm. }
\label{fig:obs:red}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{../figures/power-speckle.pdf}
\caption{Power spectra showing the influence of the \emph{post factum} reconstruction. Ordinate is the relative power on logarithmic scale, and abscissa is the spatial frequency, from the largest scales near the origin to the smallest scales at the Nyquist limit, corresponding to two pixels. A long exposure image (\emph{black dotted line}), taking the average of all speckle images, has very low noise, but the power is also low at all frequencies $\geqslant 0.8$ Mm$^{-1}$ (blurring effect). A single speckle frame (\emph{dashed blue line}) has more power at all frequencies, but also much more noise (more than two order of magnitude). The speckle reconstructed frame (\emph{red solid line}) keeps the noise low while it possesses higher power at all frequencies, where the spatial information on small-scale structures is stored.}
\label{fig:obs:speckle:power}
\end{center}
\end{figure}
\subsubsection*{Influence of the AO on the speckle interferometry\label{SIbao}}
As explained in Sec. \ref{obs:kaos} the AO systems provide a realtime correction of the low order aberrations (up to a certain order of Zernike polynomials). Nonetheless, given the anisoplanatism of the large field of view, the corrections are calculated for the lock point and applied to the whole frame, resulting in a degradation of the image correction from the lock point outwards. The problem arises from the different atmospheric columns traversed by the light from different parts in the FoV. This creates, after the AO correction, an annular dependence of the correction about the lock point and therefore an annular dependence of the $STF$s when processing the data. \cite{2006A&A...454.1011P} provided a modified version of the reconstruction code that computes different $STF$s for annular regions around the lock point, providing a more accurate treatment over the field of view.
The \emph{sigmoid} dataset was reduced using this last version of the code, improving substantially the quality of the results. Both AO and speckle interferometry work best with good seeing, and this data set was recorded under very good seeing conditions.
\subsubsection{Speckle reconstruction of the narrow-band images\label{SIn}}
The narrow-band channel scans the selected spectral line, taking several ($\sim 20$) images per spectral position. The statistical approach as for the broadband data can not be applied given the low number of frames per spectral position.
To reconstruct these images from this channel we use a method proposed by \cite{1992A&A...261..321K} and implemented in the code by \cite{2003PhDT.........2J}. For each narrow-band frame, there is a frame taken simultaneously in the broadband channel, which is degraded by the same wave aberrations. The images in the broadband channel were taken at 6300 \AA, i.e. at a wavelength 260 \AA\, shorter than that of H$\alpha$. We neglect the wavelength dependence of the aberration.
For each position in the spectral line, for each subfield, we have a set of pairs of simultaneous speckle images from the narrow- and broadband channel, with a common $OTF_{i}$ for each realization in both channels:
\begin{equation}
I_{Broad_{i}} = O_{Broad} \cdot OTF_{i}
\label{ec:obs:narrow1}
\end{equation}
\begin{equation}
I_{Narrow_{i}} = O_{Narrow} \cdot OTF_{i}
\label{ec:obs:narrow1b}
\end{equation}
Using Equation \ref{ec:obs:narrow1} in \ref{ec:obs:narrow1b}, the reconstructed narrow-band image $O_{Narrow}$ is obtained from the minimization of the error metric
\begin{equation}
E= \sum_{i=1}^{N} \Big | O_{Narrow} \cdot \frac{I_{Broad_{i}}}{O_{Broad_{i}}}-I_{Narrow_{i}} \Big |^{2} \, ,
\label{ec:obs:narrow2}
\end{equation}
where $N$ is the number of images taken at one wavelength position. Minimization of $E$ with respect to $O_{Narrow}$ yields
\begin{equation}
O_{Narrow} = H\cdot \frac{\sum_{i=1}^{N}I_{Narrow_{i}} \cdot I_{Broad_{i}}^{*}}{\sum_{i=1}^{N}|I_{Broad_{i}}|^{2}} \cdot O_{Broad_{i}} \, .
\label{ec:obs:narrow3}
\end{equation}
Here we have included a noise noise filter ($H$) to remove the power at spatial frequencies higher than a certain threshold above which the noise dominates. The noise power is obtained from the flat field data.
\subsubsection[Multi object multi frame blind deconvolution]{Multi object multi frame blind deconvolution (MOMFBD)\label{momfbd}}
The speckle interferometry method presented above relies on a statistically average influence of the wavefront aberration. In this Section we shortly present another approach that we have also used in this work. It is based on the simultaneous estimation of the object and the aberrations in a maximum likelihood sense using different simultaneous channels and several speckle frames. For more information see e.g. \citep{Lofdahl:2002qy,2005SoPh..228..191V,2007msfa.conf..119L}.
The method used is called \emph{Multi Object Multi Frame Blind Deconvolution} (MOMFBD), which historically is a modification of the ``Joint Phase Diverse Speckle'' image restoration. The original method is based on the possibility of separating the aberrations from the object if we observe simultaneously in two channels introducing a known aberration, like defocussing the image, in one of them. Mathematically, both phase diversity and multi-object methods are particularizations from the ``Multi Frame Blind Deconvolution''. Using a model of the optics, including its unknown pupil image, it is possible to jointly calculate the unaberrated object and the aberration, in a maximum likelihood sense.
Coming back to Eq. \ref{ec:obs:obser} for a single isoplanatic speckle subfield, the Optical Transfer Function (OTF) is the Fourier transform of the Point Spread Function (PSF), which is the square modulus of the Fourier transform of the pupil function (P), that can be generalized with an expression like
\begin{equation}
P= A\cdot exp(i\phi) \, ,
\label{ec:momfbd:pupil}
\end{equation}
where $A$ stands for the geometrical extent of the pupil (A$=1$ inside pupil, A$=0$ outside). This unknown phase $\phi$ can be then parametrized using a polynomial expansion:
\begin{equation}
\phi = \sum_{m\in M} \alpha_{m} \psi_{m} \, ,
\label{eq:momfdb:expan}
\end{equation}
where $\psi_{m},m \in M$, is a subset of a certain basis functions. The MOMFBD uses a combination of Zernike polynomials \citep{1976JOSA...66..207N} for tilt aberrations and Karhunen-Lo\`eve for blurring effects, as they are optimal for atmospheric blurring effects \citep{1990SPIE.1237..668R} . The $\{\alpha_{m} \}$ coefficients have therefore the information of the instantaneous wavefront aberration, whether it comes from seeing conditions, telescope aberrations or AO influence. It is interesting to note that the expansion of the phase aberration is therefore finite ($m \in M$) in our calculation, that leads to a systematic underestimation of the wings of the PSF
\citep{2005SoPh..228..191V}
For the calculation of the solution, the MOMFBD code uses a metric quantity that depends only on the $\{ \alpha_{m} \} $ parameters and is expressed as the least square difference between the $j$ speckle data frames, $D_{j} $, and the present estimated synthesized data frame, obtained by convolving the present estimation of $PSF$ and object.
\begin{equation}
%L(\alpha_{m})= \sum_{u} \Big[ \sum_{j}^{J} |D_{j}|^2 - \frac{|\sum_{j}^{J} D^{*}_{j} S_{j}|^2}{\gamma_{obj}+\sum_{j}^{J}|S_{j}|^{2}}\Big]+ \frac{\gamma_{wf}}{2}\sum_{m}^{M}\frac{1}{\lambda_{m}}\sum_{j}^{J}|\alpha_{jm}|^{2}
L(\{\alpha_{m}\})= \sum_{u} \Big[ \sum_{j}^{J} |D_{j}|^2 - \frac{|\sum_{j}^{J}O^{*}_{mj}\widehat{OTF}_{mj}|^2}{\sum_{j}^{J}|\widehat{OTF}_{mj}|^{2}+\gamma}\Big]
\end{equation}
where the $\gamma$ term accounts for the noise and corresponds to an optimum low pass filter \citep{Lofdahl:2002qy} and the $u$ index for the spatial index in the Fourier domain.
This mathematical expression, from \cite{1996ApJ...466.1087P}, to solve the blind deconvolution problem depends on the noise model used. In our case the MOMFBD assumes additive Gaussian statistics, which gives the simplest form of $L$ and the fastest code, and turns to be appropriate for low contrast objects.
The solution of the problem of image reconstruction is to find the set of $\{\alpha_{m} \}$ that minimizes the metric $L(\{\alpha_{m}\})$, providing an estimation of the OTF, and from there the new estimation of the objects. Details on the process and optimization used can be found in \cite{Lofdahl:2002qy}. The final converging solution provides thus the real object and instantaneous aberration simultaneously.
With only one channel the $\{\alpha_{m} \}$ are independent, but if we can specify linear equality constraints (LEC) to these parameters we can reduce the number of unknown coefficients for multiple channels.
The Phase Diversity method is one example of LEC. By defocussing one of the cameras on a simultaneous channel we introduce a known phase contribution in the expansion of Eq. \ref{eq:momfdb:expan}. This creates a set of related pairs of $\{\alpha_{m} \}$. Typically, 10 or even less realizations of such pairs of images are enough for a good restoration.
Different channels observing simultaneously in different, yet close, wavelengths can be used also to constrain the $\{\alpha_{m} \}$, as the instantaneous aberration can be considered the same for all channels. In our case we have several speckle images per position and two simultaneous channels. The broadband channel and the narrow-band channel scanning the spectral line at 21 positions with 20 frames per position. We have therefore a set of 21 pairs of 2 simultaneous objects, with 20 frames for each object and channel.
%?franz, better?
One interesting outcome of this multi object approach is that, if the observed data frames are previously aligned using a grid pattern, the resulting images are then perfectly aligned between simultaneous channels, which greatly reduces possible artifacts on derived quantities as Dopplergrams or magnetograms.
%?franz, better?
In this work we have used this MOMFBD approach to process the data where our usual speckle interferometry method was not applicable. This mainly applies for on-limb observations, as the limb darkening gradient on the field of view influences the statistics. Also, with the actual presence of the off-limb sky, the data are not suitable for the narrow-band speckle reconstruction, as we don't have a broadband counterpart for the emission features present off the limb.
The \emph{limb} data set was reduced using this code (see Sec. \ref{sec:limb:ha}), as well as some other data frames for comparison purposes with the speckle interferometry (Sec. \ref{sec:comp}).
The MOMFBD code was implemented by \cite{2005SoPh..228..191V} and was made freely available at \verb"www.momfbd.org". Given the high processing power needed it is written and greatly optimized in \verb"C++". It is developed to run in a multithreaded clustering environment, where the work is split in workunits and sent back from the slave machines to the master once the processing is done. A typical run with one of our H$\alpha$ scans in broad and narrow-band channel, reconstructing the first 50 Karhunen-Lo\`eve modes, takes $\sim7$ hours to process with 20 CPU cores of $3.2$ GHz.
\section{Infrared spectrometry}
For this work we have also used spectroscopic data in the infrared region, to study the spicular emission in the \ion{He}{i} 10830 \AA\, multiplet.
%franz, on introduction i talk about the spectral line, origin, levels...
For this purpose we used the echelle spectrograph of the VTT and the Tenerife Infrared Polarimeter (TIP).
In this Section we summarize the instrument characteristics, the optical setup
%optical setup is the figure 2.1 basically
and the observations performed for the study of the emission profiles of spicules, which will be presented in Chapter \ref{ch:spicules}.
\subsection{Instrument\label{inst:tip}}
TIP was developed at the Instituto de Astrof\'isica de Canarias \citep{Martinez-Pillet:1999lr} and recently upgraded with a new, larger infrared CCD detector \citep{Collados:2007fk}. It is able to record simultaneously all four Stokes components with very high spectral resolution in the infrared region from $1 \mu m $ to $2.3 \mu m$, with a fast cadence and very high spatial resolution along the slit.
The optical setup of the instrument is shown in Fig. \ref{fig:tip:optical}. After the main tank and the AO system, a narrow ($\sim100 \,\mu$m wide) slit is mounted in the plane of the prime focus of the telescope. The light reflected from the slit jaws enters a camera system to provide images, to point the telescope and to have the region of interest imaged onto the slit. The small fraction of light entering the slit goes through the polarimeter, where the Stokes components are modulated. Then, the predisperser and spectrograph decompose the light into its spectral components. At the end of the optical path the detector is mounted, a CCD cooled below 100 K
%Franz check temperature of the TIP camera
to reduce the thermal excitation of electrons in the CCD pixels.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{../figures/tip-opt.png}
\caption{Optical schema of the Tenerife Infrared Polarimeter (TIP) with slit jaw camera, predisperser and spectrograph of the VTT. After the AO correction, the light from the prime focus of the telescope enters the instrument through the slit. The light reflected from the slit jaws is recorded with video cameras to create context frames. After the slit, the polarimeter with the ferroelectric liquid crystals modulates the polarization of the light beam. The predisperser selects, with mask (p1), the spectral region to observe, and the spectrograph disperses the light into its spectral components. The nitrogen-cooled CCD detector records the modulated polarization of the spectra. d1 and d2 are the diffraction gratings.}
\label{fig:tip:optical}
\end{center}
\end{figure}
\subsubsection*{The polarimeter\label{polarimeter}}
TIP is able to obtain simultaneously the full set of the four Stokes parameters that determines the polarization of the light, from each point in the slit. However, this work concentrates only on the intensity measurements. The polarization measurement is performed by means of two ferroelectric liquid crystals (FLC). These are electro-optic materials with fixed optical retardation, whose axis can be switched between two orientations by applying voltages of approximately $\pm$ 10V. This amplitude of the rotation of the retardation axis is somewhat dependent on the temperature, and is $\sim 45^{\circ}$ at $20-25$C. With two FLCs, with two possible states each, we can create four different combinations of modulation of the incident light. The four modulated intensities are four different linear combinations of \{I,Q,U,V\} with different weights on each parameter. With four consecutive measurements we can therefore retrieve the four components of the Stokes vector. Thus, TIP is able to obtain simultaneously the four components of the polarization for each full cycle of the polarimeter. Although TIP makes a full cycle of the FLCs in less than one second, we have to accumulate several spectrograms in order to increase the signal to noise ratio, especially when measuring weak signals like the polarization of spicules outside the solar limb.
In the sequence following the light path, the physical setup of the polarimeter consists of a UV-blocking filter to protect the FLCs from intense high energy radiation at short wavelength. Then, the first FLC with a retardation of $\lambda/2$ and the second FLC with $\lambda/4$ follow. The retardances of $\lambda/2$ and $\lambda/4$ are nominal values. The actual retardances differ from these values and depend on wavelength. Finally a Savart plate splits the light into two orthogonal linearly polarized beams.
As part of the instruments we need a calibration optic subsystem (see explanation in Sec. \ref{tip:reduc}) to account for the influence of the mirrors following the telescope. For this reason, in front of the AO system, there is a polarization calibration unit (PCU) that can be moved into the light path. It is composed of a retarder with nominal retardance of $\lambda/4$ in the optical spectral range, and a fixed linear polarizer. The retarder rotates a full cycle with measurements taken every 5 degrees, creating a set of 73 modulations of the light beam that are used to model the influence of the optics behind the telescope, but including AO, till the detector. The influence of the coelostat mirrors and the telescope proper on the polarization state are taken into account with a polarization model of these parts by \cite{2005A&A...443.1047B}.
\subsection{Observations\label{obs:tip}}
Table \ref{table:obs:tip} summarizes the details of the observing campaign for the course of this work. It focuses on studying the emission profiles observed in spicules in the \ion{He}{i} 10830 \AA\ multiplet.
The strong darkening close to the solar limb and the presence of the
limb make it difficult to use KAOS for off-limb observations, since the
correlation algorithm of KAOS was not developed for this kind of observations.
We scanned the full height of the spicule extension, starting inside the disc. We made a single spatial scan with long integration time per position. As the \emph{lock point} of the AO was placed on a nearby facula inside the disc was chosen. Apart from the facula used for AO tracking, it was a quiet Sun region. In the present work we study only the intensity component of the Stokes vector \citep[see definition in e.g.][]{Chandrasekhar:1960lr,wikistokes}.
\begin{table}[t]
\begin{center}\begin{tabular}{|r|c|c|c|}\hline
%\textbf{ Dataset name} & \textit{``intensity''} & \textit{``photometric''} & \textit{``polarimetric''} \\\hline
\hline
Date & Dec,4$^{th} $,2005 %& Oct,14$^{th} $,2006 & May,20$^{th}$, 2007
\\\hline
% Type of data & Intensity & I (photometry mode) & full Stokes \\\hline
Location & NE limb % & S limb & E limb
\\\hline
Spectral sampling \# & 10.9 m\AA/px %\multicolumn{3}{|c|}{10.9 m\AA/px}
\\\hline
Time span & 1 scan in 66 min. % & 7 scans in 19.6 min. & 4 scans in 23 min
\\\hline
Slit & 40\arcsec $ \times $ 0\farcs5 %&$\sim$80\arcsec$ \times $0\farcs67 &$\sim$ 40\arcsec$ \times $0\farcs5
\\\hline
Integration time & 5$ \times $2.5 s %& \multicolumn{2}{|c|}{3 s}
\\\hline
Step size & 0\farcs35 %& \multicolumn{2}{|c|}{0\farcs5}
\\\hline
Max. height off-limb & 7\arcsec % & \multicolumn{2}{|c|}{$\sim$13 \arcsec}
\\\hline
Seeing condition ($r_{0}$) & $\sim7$cm (max 12 cm) %& $\sim5.5$cm (max 8 cm) & $\sim8$cm (max 12 cm)
\\\hline
KAOS support & yes %\multicolumn{2}{|c|}{yes} & yes (improved)
\\\hline
\end{tabular} \caption{Characteristics of the data taken with TIP used in this work. $r_{0}$ is the Fried parameter.\label{table:obs:tip}}
\end{center}
\end{table}
\subsection{Data reduction\label{tip:reduc}}
As for the G-FPI case, the data reduction process aims to remove the instrumental effects as well as the atmospheric influence. For TIP data this involves three steps. The first is common to all CCD observations and consists in removing instrumental effects, the second is the polarimetric calibration of the signal, and the third is the spectrosposcopic calibration.
\subsubsection*{Reduction of CCD effects}
\begin{figure}[t]
\centering
\subfloat[Frame of raw data]{
\includegraphics[width=0.45\textwidth]{../figures/rawtip.jpg}}%
\quad%
\subfloat[Flat field ]{
\includegraphics[width=0.45\textwidth]{../figures/flattip.jpg}}
\\
\quad%
\subfloat[Dark frame ]{
\includegraphics[width=0.45\textwidth]{../figures/darktip.jpg}}
\quad%
\subfloat[Reduced frame]{
\includegraphics[width=0.45\textwidth]{../figures/redtip.jpg}}
\caption{Examples of the standard data reduction process for spectral data. The Flat field frame (b) is calculated dividing average flat field data by the mean spectra of the average. %These example frames correspond to the \emph{photometric} data set, with doubled slit length.
}
\label{tip:flat}
\end{figure}
This processing is basically the same for all CCD observations: removal of dark counts and correction for differential sensitivity of the pixel matrix with the gain table (using the flat fields). The only difference to G-FPI data reduction is when creating the flat fields. The mean flat field frame is not \emph{flat}. Although being a spatial average, it still contains spectral information. To retain only the gain table information we divide the flat field by the mean spectrogram, so that only the differential response of the pixels is left (see Fig. \ref{tip:flat}). The mean spectrogram is obtained by averaging the flat field spectrograms over the spatial coordinate.
\subsubsection*{Polarimetric calibration}
The signals recorded with the CCD are not directly the Stokes parameters \cite[see description in e.g. ][]{Chandrasekhar:1960lr} . With two FLCs we have four different combinations in one full cycle. For each configuration in the cycle, we measure intensities as a particular linear combination of \{I,Q,U,V\} with different weights, so we can solve the ensuing system of equations. Also, in each CCD frame, we measure light of two orthogonal linearly polarized beams (see Sec. \ref{inst:tip}).
An important problem in polarimetric observations is that each reflecting surface of the telescope changes the polarization state of the incoming light. So the optical path, with all the reflecting surfaces from the coelostat to the CCD, introduces a complex modulation of the incoming polarization. At the VTT there is a polarization calibration unit (PCU) mounted in front of the AO system. This device feeds the subsequent optical components with light of well defined polarization states. So, once we have a set of Stokes parameters from different configurations of the PCU, we can obtain the modulation induced by the optical path, the Mueller matrix $ \mathbb{M}$, from the PCU to the polarimeter:
\begin{equation}
\left(\begin{array}{c}I \\Q \\U\\V\end{array}\right)_{polarimeter} = \mathbb{M} \cdot \left(\begin{array}{c}I \\Q \\U\\V\end{array}\right)_{input}
\end{equation}
The inverse matrix of $\mathbb{M}$ will therefore relate the polarization state of the light that reaches the polarimeter with the light arriving at the PCU position. However, the light path from the coelostat to the PCU (in front of the AO) cannot be calibrated with this system, so the reduction routines use a theoretical model of this part of the telescope.
This process is already implemented with available reduction pipelines. Further investigation of \emph{crosstalk} or other additional polarimetric reduction are needed to reduce the instrumental effect in our data. However, this is not necessary for our case, since this work concentrates only on the intensity component.
\begin{comment}
When one of the components of the Stokes vector is mixed into the others (for example I into Q, U or V) we have a contamination of those components, or \emph{crosstalk}. After the polarimetric reduction process there can still be some residual \emph{crosstalk} which can be removed using statistical methods \citep{Collados:2003lr} . Unfortunately this method is not appropriate for our limb observations, since it is based on assumptions which are valid only near disc center. In our case we remove the residual \emph{crosstalk} using the following methods (all other residual \emph{crosstalks}, like V $\rightarrow$ Q, are an order of magnitude lower and are not treated):
\begin{itemize}
\item $I_{disc}$ $\rightarrow$ \{Q,U,V\} : Although we are observing off the limb, at small distances to the disc there is an important contribution of the disc spectra to the data, due to the scattering by the atmosphere and due to image motion and blurring.
This spectral intensity profile of unpolarized light from the disc can also contaminate the other Stokes profiles. The observed spectral region contains, apart from spectral lines, also continua. Since the polarization in these continuum regions should be zero, all non-zero polarization must come from the contamination of I, so we know the strength of the disc signal that should be subtracted.
\item $I_{off-the-limb}$ $\rightarrow$ \{Q,U,V\} : The intensity signal of the emission profiles can also produce false signals in other Stokes components. To remove them (wherever detected), we use the fact that the blue component of the \ion{He}{i} 10830 \AA\ multiplet is not polarized and therefore should not show any Q, U or V.
Since we know that Q and U both should be symmetric and V antisymmetric, we can therefore estimate the crosstalk from I.
\end{itemize}
Finally rotate the axis which defines the orientation of linear polarization (see Sec. \ref{intro:polarimetry}). We want to have $Q > 0$ parallel to the limb. Given the definition of the Stokes parameters this transformation is simply:
\begin{eqnarray}
Q_{limb} = \cos(2\alpha) \cdot Q_{N} + \sin(2\alpha) \cdot U_{N}
\cr
U_{limb} = -\sin(2\alpha) \cdot Q_{N} + \cos(2\alpha) \cdot U_{N}
\end{eqnarray}
where $\alpha$ is the angle between the observed limb and terrestrial north-south direction (subscript N). This latter direction is the conventional one according the instrument calibration process.
\end{comment}
\subsubsection*{Spectroscopic reduction}
The last type of reduction procedure is related to the nature of spectroscopic data and consists of the calibration in wavelength, the continuum correction and a low pass filtering to remove noise.
To calibrate our spectrograms in wavelength we make use of the two telluric lines present in our spectral range of the TIP data. Solar lines are subject to Doppler shifts from local flows and solar rotation. Yet, telluric absorption lines are formed in the atmosphere of the Earth. Therefore, they are always narrow due to only small Doppler broadening and are located at fixed wavelength. This provides a fixed reference coordinate that we use with the FTS atlas \citep{Neckel:1999lr}. Comparing both spectra we can accurately measure the spectral sampling which is for all data sets $10.9 $m\AA/pixel\,. See wavelength scale abscissa of Fig. \ref{fig:tip:cont}.
The transmission of the filters is not a constant in the transmitted wavelength range, so this creates an intensity variation curve in all our spectrograms. For normalization, we have to find the correct level of the continuum intensities of the spectrograms observed on the disc. For this, we use several spectral positions between spectral lines and calculate the ratio between the observed data and the values from the FTS atlas. We interpolate to create the continuum correction (see green dashed line on Fig. \ref{fig:tip:cont}).