#  Published in Astroparticle Physics as
DOI:10.1016/j.astropartphys.2017.09.001 Spectral Calibration of the
Fluorescence Telescopes of the Pierre Auger Observatory

A. Aab  P. Abreu  M. Aglietta  I. Al Samarai  I.F.M. Albuquerque  I. Allekotte
A. Almela  J. Alvarez Castillo  J. Alvarez-Muñiz  G.A. Anastasi  L.
Anchordoqui  B. Andrada  S. Andringa  C. Aramo  F. Arqueros  N. Arsene  H.
Asorey  P. Assis  J. Aublin  G. Avila  A.M. Badescu  A. Balaceanu  F. Barbato
R.J. Barreira Luz  J.J. Beatty  K.H. Becker  J.A. Bellido  C. Berat  M.E.
Bertaina  P.L. Biermann  J. Biteau  S.G. Blaess  A. Blanco  J. Blazek  C.
Bleve  M. Boháčová  D. Boncioli  C. Bonifazi  N. Borodai  A.M. Botti  J. Brack
I. Brancus  T. Bretz  A. Bridgeman  F.L. Briechle  P. Buchholz  A. Bueno  S.
Buitink  M. Buscemi  K.S. Caballero-Mora  L. Caccianiga  A. Cancio  F. Canfora
L. Caramete  R. Caruso  A. Castellina  F. Catalani  G. Cataldi  L. Cazon  A.G.
Chavez  J.A. Chinellato  J. Chudoba  R.W. Clay  A. Cobos  R. Colalillo  A.
Coleman  L. Collica  M.R. Coluccia  R. Conceição  G. Consolati  F. Contreras
M.J. Cooper  S. Coutu  C.E. Covault  J. Cronin  S. D’Amico  B. Daniel  S.
Dasso  K. Daumiller  B.R. Dawson  R.M. de Almeida  S.J. de Jong  G. De Mauro
J.R.T. de Mello Neto  I. De Mitri  J. de Oliveira  V. de Souza  J. Debatin  O.
Deligny  M.L. Díaz Castro  F. Diogo  C. Dobrigkeit  J.C. D’Olivo  Q. Dorosti
R.C. dos Anjos  M.T. Dova  A. Dundovic  J. Ebr  R. Engel  M. Erdmann  M.
Erfani  C.O. Escobar  J. Espadanal  A. Etchegoyen  H. Falcke  J. Farmer  G.
Farrar  A.C. Fauth  N. Fazzini  F. Fenu  B. Fick  J.M. Figueira  A. Filipčič
O. Fratu  M.M. Freire  T. Fujii  A. Fuster  R. Gaior  B. García  D. Garcia-
Pinto  F. Gaté  H. Gemmeke  A. Gherghel-Lascu  P.L. Ghia  U. Giaccari  M.
Giammarchi  M. Giller  D. Głas  C. Glaser  G. Golup  M. Gómez Berisso  P.F.
Gómez Vitale  N. González  B. Gookin  A. Gorgi  P. Gorham  A.F. Grillo  T.D.
Grubb  F. Guarino  G.P. Guedes  R. Halliday  M.R. Hampel  P. Hansen  D. Harari
T.A. Harrison  J.L. Harton  A. Haungs  T. Hebbeker  D. Heck  P. Heimann  A.E.
Herve  G.C. Hill  C. Hojvat  E. Holt  P. Homola  J.R. Hörandel  P. Horvath  M.
Hrabovský  T. Huege  J. Hulsman  A. Insolia  P.G. Isar  I. Jandt  J.A. Johnsen
M. Josebachuili  J. Jurysek  A. Kääpä  O. Kambeitz  K.H. Kampert  B. Keilhauer
N. Kemmerich  E. Kemp  J. Kemp  R.M. Kieckhafer  H.O. Klages  M. Kleifges  J.
Kleinfeller  R. Krause  N. Krohm  D. Kuempel  G. Kukec Mezek  N. Kunka  A.
Kuotb Awad  B.L. Lago  D. LaHurd  R.G. Lang  M. Lauscher  R. Legumina  M.A.
Leigui de Oliveira  A. Letessier-Selvon  I. Lhenry-Yvon  K. Link  D. Lo Presti
L. Lopes  R. López  A. López Casado  R. Lorek  Q. Luce  A. Lucero  M. Malacari
M. Mallamaci  D. Mandat  P. Mantsch  A.G. Mariazzi  I.C. Mariş  G. Marsella
D. Martello  H. Martinez  O. Martínez Bravo  J.J. Masías Meza  H.J. Mathes  S.
Mathys  J. Matthews  J.A.J. Matthews  G. Matthiae  E. Mayotte  P.O. Mazur  C.
Medina  G. Medina-Tanco  D. Melo  A. Menshikov  K.-D. Merenda  S. Michal  M.I.
Micheletti  L. Middendorf  L. Miramonti  B. Mitrica  D. Mockler  S. Mollerach
F. Montanet  C. Morello  M. Mostafá  A.L. Müller  G. Müller  M.A. Muller  S.
Müller  R. Mussa  I. Naranjo  L. Nellen  P.H. Nguyen  M. Niculescu-Oglinzanu
M. Niechciol  L. Niemietz  T. Niggemann  D. Nitz  D. Nosek  V. Novotny  L.
Nožka  L.A. Núñez  L. Ochilo  F. Oikonomou  A. Olinto  M. Palatka  J. Pallotta
P. Papenbreer  G. Parente  A. Parra  T. Paul  M. Pech  F. Pedreira  J. Pȩkala
R. Pelayo  J. Peña-Rodriguez  L. A. S. Pereira  M. Perlin  L. Perrone  C.
Peters  S. Petrera  J. Phuntsok  R. Piegaia  T. Pierog  M. Pimenta  V.
Pirronello  M. Platino  M. Plum  C. Porowski  R.R. Prado  P. Privitera  M.
Prouza  E.J. Quel  S. Querchfeld  S. Quinn  R. Ramos-Pollan  J. Rautenberg  D.
Ravignani  J. Ridky  F. Riehn  M. Risse  P. Ristori  V. Rizi  W. Rodrigues de
Carvalho  G. Rodriguez Fernandez  J. Rodriguez Rojo  D. Rogozin  M.J.
Roncoroni  M. Roth  E. Roulet  A.C. Rovero  P. Ruehl  S.J. Saffi  A. Saftoiu
F. Salamida  H. Salazar  A. Saleh  F. Salesa Greus  G. Salina  F. Sánchez  P.
Sanchez-Lucas  E.M. Santos  E. Santos  F. Sarazin  R. Sarmento  C. Sarmiento-
Cano  R. Sato  M. Schauer  V. Scherini  H. Schieler  M. Schimp  D. Schmidt  O.
Scholten  P. Schovánek  F.G. Schröder  S. Schröder  A. Schulz  J. Schumacher
S.J. Sciutto  A. Segreto  A. Shadkam  R.C. Shellard  G. Sigl  G. Silli  O.
Sima  A. Śmiałkowski  R. Šmída  G.R. Snow  P. Sommers  S. Sonntag  R.
Squartini  D. Stanca  S. Stanič  J. Stasielak  P. Stassi  M. Stolpovskiy  F.
Strafella  A. Streich  F. Suarez  M. Suarez Durán  T. Sudholz  T. Suomijärvi
A.D. Supanitsky  J. Šupík  J. Swain  Z. Szadkowski  A. Taboada  O.A. Taborda
V.M. Theodoro  C. Timmermans  C.J. Todero Peixoto  L. Tomankova  B. Tomé  G.
Torralba Elipe  P. Travnicek  M. Trini  R. Ulrich  M. Unger  M. Urban  J.F.
Valdés Galicia  I. Valiño  L. Valore  G. van Aar  P. van Bodegom  A.M. van den
Berg  A. van Vliet  E. Varela  B. Vargas Cárdenas  G. Varner  R.A. Vázquez  D.
Veberič  C. Ventura  I.D. Vergara Quispe  V. Verzi  J. Vicha  L. Villaseñor
S. Vorobiov  H. Wahlberg  O. Wainberg  D. Walz  A.A. Watson  M. Weber  A.
Weindl  L. Wiencke  H. Wilczyński  M. Wirtz  D. Wittkowski  B. Wundheiler  L.
Yang  A. Yushkov  E. Zas  D. Zavrtanik  M. Zavrtanik  A. Zepeda  B. Zimmermann
M. Ziolkowski  Z. Zong  F. Zuccarello  The Pierre Auger Collaboration  Centro
Atómico Bariloche and Instituto Balseiro (CNEA-UNCuyo-CONICET), San Carlos de
Bariloche, Argentina  Centro de Investigaciones en Láseres y Aplicaciones,
CITEDEF and CONICET, Villa Martelli, Argentina  Departamento de Física and
Departamento de Ciencias de la Atmósfera y los Océanos, FCEyN, Universidad de
Buenos Aires and CONICET, Buenos Aires, Argentina  IFLP, Universidad Nacional
de La Plata and CONICET, La Plata, Argentina  Instituto de Astronomía y Física
del Espacio (IAFE, CONICET-UBA), Buenos Aires, Argentina  Instituto de Física
de Rosario (IFIR) – CONICET/U.N.R. and Facultad de Ciencias Bioquímicas y
Farmacéuticas U.N.R., Rosario, Argentina  Instituto de Tecnologías en
Detección y Astropartículas (CNEA, CONICET, UNSAM), and Universidad
Tecnológica Nacional – Facultad Regional Mendoza (CONICET/CNEA), Mendoza,
Argentina  Instituto de Tecnologías en Detección y Astropartículas (CNEA,
CONICET, UNSAM), Buenos Aires, Argentina  Observatorio Pierre Auger, Malargüe,
Argentina  Observatorio Pierre Auger and Comisión Nacional de Energía Atómica,
Malargüe, Argentina  Universidad Tecnológica Nacional – Facultad Regional
Buenos Aires, Buenos Aires, Argentina  University of Adelaide, Adelaide, S.A.,
Australia  Université Libre de Bruxelles (ULB), Brussels, Belgium  Centro
Brasileiro de Pesquisas Fisicas, Rio de Janeiro, RJ, Brazil  Centro Federal de
Educação Tecnológica Celso Suckow da Fonseca, Nova Friburgo, Brazil
Universidade de São Paulo, Escola de Engenharia de Lorena, Lorena, SP, Brazil
Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP,
Brazil  Universidade de São Paulo, Instituto de Física, São Paulo, SP, Brazil
Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil  Universidade
Estadual de Feira de Santana, Feira de Santana, Brazil  Universidade Federal
de Pelotas, Pelotas, RS, Brazil  Universidade Federal do ABC, Santo André, SP,
Brazil  Universidade Federal do Paraná, Setor Palotina, Palotina, Brazil
Universidade Federal do Rio de Janeiro, Instituto de Física, Rio de Janeiro,
RJ, Brazil  Universidade Federal do Rio de Janeiro (UFRJ), Observatório do
Valongo, Rio de Janeiro, RJ, Brazil  Universidade Federal Fluminense, EEIMVR,
Volta Redonda, RJ, Brazil  Universidad Industrial de Santander, Bucaramanga,
Colombia  Charles University, Faculty of Mathematics and Physics, Institute of
Particle and Nuclear Physics, Prague, Czech Republic  Institute of Physics of
the Czech Academy of Sciences, Prague, Czech Republic  Palacky University,
RCPTM, Olomouc, Czech Republic  Institut de Physique Nucléaire d’Orsay (IPNO),
Université Paris-Sud, Univ. Paris/Saclay, CNRS-IN2P3, Orsay, France
Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Universités
Paris 6 et Paris 7, CNRS-IN2P3, Paris, France  Laboratoire de Physique
Subatomique et de Cosmologie (LPSC), Université Grenoble-Alpes, CNRS/IN2P3,
Grenoble, France  Bergische Universität Wuppertal, Department of Physics,
Wuppertal, Germany  Karlsruhe Institute of Technology, Institut für
Experimentelle Kernphysik (IEKP), Karlsruhe, Germany  Karlsruhe Institute of
Technology, Institut für Kernphysik, Karlsruhe, Germany  Karlsruhe Institute
of Technology, Institut für Prozessdatenverarbeitung und Elektronik,
Karlsruhe, Germany  RWTH Aachen University, III. Physikalisches Institut A,
Aachen, Germany  Universität Hamburg, II. Institut für Theoretische Physik,
Hamburg, Germany  Universität Siegen, Fachbereich 7 Physik – Experimentelle
Teilchenphysik, Siegen, Germany  Gran Sasso Science Institute (INFN),
L’Aquila, Italy  INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica di
Palermo, Palermo, Italy  INFN Laboratori Nazionali del Gran Sasso, Assergi
(L’Aquila), Italy  INFN, Sezione di Catania, Catania, Italy  INFN, Sezione di
Lecce, Lecce, Italy  INFN, Sezione di Milano, Milano, Italy  INFN, Sezione di
Napoli, Napoli, Italy  INFN, Sezione di Roma ”Tor Vergata”, Roma, Italy  INFN,
Sezione di Torino, Torino, Italy  Osservatorio Astrofisico di Torino (INAF),
Torino, Italy  Politecnico di Milano, Dipartimento di Scienze e Tecnologie
Aerospaziali , Milano, Italy  Università del Salento, Dipartimento di
Ingegneria, Lecce, Italy  Università del Salento, Dipartimento di Matematica e
Fisica “E. De Giorgi”, Lecce, Italy  Università dell’Aquila, Dipartimento di
Scienze Fisiche e Chimiche, L’Aquila, Italy  Università di Catania,
Dipartimento di Fisica e Astronomia, Catania, Italy  Università di Milano,
Dipartimento di Fisica, Milano, Italy  Università di Napoli ”Federico II”,
Dipartimento di Fisica “Ettore Pancini“, Napoli, Italy  Università di Roma
“Tor Vergata”, Dipartimento di Fisica, Roma, Italy  Università Torino,
Dipartimento di Fisica, Torino, Italy  Benemérita Universidad Autónoma de
Puebla, Puebla, México  Centro de Investigación y de Estudios Avanzados del
IPN (CINVESTAV), México, D.F., México  Unidad Profesional Interdisciplinaria
en Ingeniería y Tecnologías Avanzadas del Instituto Politécnico Nacional
(UPIITA-IPN), México, D.F., México  Universidad Autónoma de Chiapas, Tuxtla
Gutiérrez, Chiapas, México  Universidad Michoacana de San Nicolás de Hidalgo,
Morelia, Michoacán, México  Universidad Nacional Autónoma de México, México,
D.F., México  Institute of Nuclear Physics PAN, Krakow, Poland  University of
Łódź, Faculty of Astrophysics, Łódź, Poland  University of Łódź, Faculty of
High-Energy Astrophysics,Łódź, Poland  Laboratório de Instrumentação e Física
Experimental de Partículas – LIP and Instituto Superior Técnico – IST,
Universidade de Lisboa – UL, Lisboa, Portugal  “Horia Hulubei” National
Institute for Physics and Nuclear Engineering, Bucharest-Magurele, Romania
Institute of Space Science, Bucharest-Magurele, Romania  University
Politehnica of Bucharest, Bucharest, Romania  Experimental Particle Physics
Department, J. Stefan Institute, Ljubljana, Slovenia  Center for Astrophysics
and Cosmology (CAC), University of Nova Gorica, Nova Gorica, Slovenia
Universidad Complutense de Madrid, Madrid, Spain  Universidad de Granada and
C.A.F.P.E., Granada, Spain  Universidad de Santiago de Compostela, Santiago de
Compostela, Spain  IMAPP, Radboud University Nijmegen, Nijmegen, The
Netherlands  KVI – Center for Advanced Radiation Technology, University of
Groningen, Groningen, The Netherlands  Nationaal Instituut voor Kernfysica en
Hoge Energie Fysica (NIKHEF), Science Park, Amsterdam, The Netherlands
Stichting Astronomisch Onderzoek in Nederland (ASTRON), Dwingeloo, The
Netherlands  Case Western Reserve University, Cleveland, OH, USA  Colorado
School of Mines, Golden, CO, USA  Department of Physics and Astronomy, Lehman
College, City University of New York, Bronx, NY, USA  Louisiana State
University, Baton Rouge, LA, USA  Michigan Technological University, Houghton,
MI, USA  New York University, New York, NY, USA  Northeastern University,
Boston, MA, USA  Ohio State University, Columbus, OH, USA  Pennsylvania State
University, University Park, PA, USA  University of Chicago, Enrico Fermi
Institute, Chicago, IL, USA  University of Nebraska, Lincoln, NE, USA  School
of Physics and Astronomy, University of Leeds, Leeds, United Kingdom  Max-
Planck-Institut für Radioastronomie, Bonn, Germany  also at Vrije Universiteit
Brussels, Brussels, Belgium  now at Deutsches Elektronen-Synchrotron (DESY),
Zeuthen, Germany  SUBATECH, École des Mines de Nantes, CNRS-IN2P3, Université
de Nantes, France  Fermi National Accelerator Laboratory, USA  University of
Bucharest, Physics Department, Bucharest, Romania  Colorado State University,
Fort Collins, CO  University of Hawaii, Honolulu, HI, USA  University of New
Mexico, Albuquerque, NM, USA

###### Abstract

We present a novel method to measure precisely the relative spectral response
of the fluorescence telescopes of the Pierre Auger Observatory. We used a
portable light source based on a xenon flasher and a monochromator to measure
the relative spectral efficiencies of eight telescopes in steps of 5 nm from
280 nm to 440 nm. Each point in a scan had approximately 2 nm FWHM out of the
monochromator. Different sets of telescopes in the observatory have different
optical components, and the eight telescopes measured represent two each of
the four combinations of components represented in the observatory. We made an
end-to-end measurement of the response from different combinations of optical
components, and the monochromator setup allowed for more precise and complete
measurements than our previous multi-wavelength calibrations. We find an
overall uncertainty in the calibration of the spectral response of most of the
telescopes of 1.5% for all wavelengths; the six oldest telescopes have larger
overall uncertainties of about 2.2%. We also report changes in physics
measureables due to the change in calibration, which are generally small.

###### keywords:

Auger Observatory , Nitrogen Fluorescence , Extensive Air Shower , Calibration

##  1 Introduction

The Pierre Auger Observatory [1] has been designed to study the origin and the
nature of ultra high-energy cosmic rays, which have energies above $10^{18}$
eV. The construction of the complete observatory following the original design
finished in 2008. The observatory is located in Malargüe, Argentina, and
consists of two complementary detector systems, which provide independent
information on the cosmic ray events. Extensive Air Showers (EAS) initiated by
cosmic rays in the Earth’s atmosphere are measured by the Surface Detector
(SD) and the Fluorescence Detector (FD). The SD is composed of 1660 water
Cherenkov detectors located mostly on a triangular array of 1.5 km spacing
covering an area of roughly 3000 km2. The SD measures the EAS secondary
particles reaching ground level [2]. The FD is designed to measure the
nitrogen fluorescence light produced in the atmosphere by the EAS secondary
particles. The FD is composed of 27 telescopes overlooking the SD array from
four sites, Los Leones (LL), Los Morados (LM), Loma Amarilla (LA), and
Coihueco (CO) [3]. The SD takes data continuously, but the FD operates only on
clear nights, and care is taken to avoid exposure to too much moonlight.

The energy of the primary cosmic ray is a key measurable for the science of
the observatory, and the FD measurement of the energy, with lower independent
systematic uncertainties, is used to calibrate the SD energy scale using
events observed by both detectors. The work described here explains how the FD
calibration at wavelengths across the nitrogen fluorescence spectrum has
recently been improved, resulting in smaller related systematic uncertainties.

The buildings at the four FD sites each have six independent telescopes, and
each telescope has a 30°$\times$ 30° field of view, leading to a 180° coverage
in azimuth and from 2° to 32° in elevation at each building. Additionally,
three specialized telescopes called HEAT [4] are located near Coihueco to
overlook a portion of the SD array at higher elevations, from 32° to 62°, to
register EAS of lower energies. All these telescopes are housed in climate-
controlled buildings, isolated from dust and day light. The layout of the
observatory is shown schematically in Figure 1.

Figure 1: A schematic of the Pierre Auger Observatory where each black dot is
a water Cherenkov detector. Locations of the fluorescence telescopes are shown
along the perimeter of the surface detector array, where the blue lines
indicate their individual field of view. The field of view of the HEAT
telescopes are indicated with red lines.

Each FD telescope is composed of several optical components as shown in Figure
2: a 2.2 m aperture diaphragm, a UV filter to reduce the background light, a
Schmidt corrector annulus, a 3.5 m $\times$ 3.5 m tessellated spherical
mirror, and a camera formed by an array of 440 hexagonal photomultipliers
(PMT) each with a field of view of 1.5° full angle. Each PMT has a light
concentrator approximating a hexagonal Winston cone to reduce dead spaces
between PMTs [3].

Figure 2: The optical components of an individual fluorescence telescope.

The energy calibration of the data [5, 6] for the Pierre Auger Observatory,
including events observed by the SD only, relies on the calibration of the FD.
Events observed by both FD and SD provide the link from the FD, which is
absolutely calibrated, to the SD data. To calibrate the FD three different
procedures are performed: the absolute [7], the relative [8], and the spectral
(or multi-wavelength) calibrations [9]. We focus here on the spectral
calibration, which is a relative measurement that relates the absolute
calibration performed at 365 nm to wavelengths across the nitrogen
fluorescence spectrum, which is shown in Figure 3.

Figure 3: The nitrogen fluorescence spectrum as measured by the AIRFLY
collaboration [10] showing the 21 major transitions.

To perform this measurement the drum-shaped portable light source used for the
absolute calibration [7] was adapted to emit UV light across the wavelength
range of interest. The drum light source is designed to uniformly illuminate
all 440 PMTs in a single camera simultaneously when it is placed at the
aperture of the FD telescope, enabling the end-to-end calibration.

The FD response as a function of wavelength was initially calculated as a
convolution of separate reflection or transmission measurements of each
optical component used in the first Los Leones telescopes [11]. The first end-
to-end spectral calibration of the FD was performed using the drum light
source with a xenon flasher and filter wheel to provide five points across the
FD wavelength response [9]. This measurement represented an improvement for
the energy estimation of all events observed by the Pierre Auger Observatory
as it has been shown to increase the reconstructed energy of events by nearly
4% for all energies [12]. However, that result has two limitations: first, the
differences in FD optical components were not measured since only one
telescope was calibrated; and second, determining the FD spectral response
curve using only five points involved a complicated fitting procedure, and was
particularly difficult considering the large width of the filters, which
resulted in relatively large systematic uncertainties.

The aim of the work described in this paper was to measure the FD efficiency
at many points across the nitrogen fluorescence spectrum with a reduced
wavelength bite at each point, and to do it at enough telescopes to cover the
different combinations of optical components making up all the telescopes
within the Auger Observatory. The spectral calibration described here proceeds
in three steps. First, the relative drum emission spectrum is measured in the
dark hall lab in Malargüe with a specific calibration PMT, called the “Lab-
PMT”, observing the drum at a large distance, in a similar fashion to the
absolute calibration of the drum; see [7] and explanatory drawings therein.
Knowing the intensity of the drum at each wavelength, we next measure the
response of the FD telescopes to the output of the multi-wavelength drum over
the course of several nights, while recording data from a monitoring
photodiode (PD) exposed to the narrow-band light at each point to ensure
knowledge of the relative drum spectrum. Finally, the FD telescope response is
normalized by the measured relative drum emission spectrum at every
wavelength, and we evaluate the associated systematic uncertainties in the
final calculation of the efficiency.

This following sections describe the measurements and analysis of data taken
during March 2014: FD optical components in section 2; the new drum light
source in section 3; measurements of the drum light source spectrum in section
4; calibrations performed at the FD telescopes in section 5; FD efficiency as
a function of wavelength in section 6; and final calibration results in
section 7. Effects on physics measurables due to changing calibrations are
discussed in section 8.

##  2 Optical Components of the Fluorescence Telescopes

There are two types of mirrors used in the telescopes, and the glass used for
the corrector rings was produced using two different glass-making procedures.
The 12 mirrors at Los Leones and Los Morados are aluminum with a 2 mm AlMgSiO5
layer glued on as the reflective surface, and the 12 mirrors at Coihueco and
Loma Amarilla are composed of a borosilicate glass with a 90 nm Al layer and
then a 110 nm SiO2 layer (see [3] for more details). Two different procedures
were used to grow the borosilicate glass used in the corrector rings, both by
Schott Glass Manufactures111Schott Glass,
http://www.us.schott.com/english/index.html. One type is called Borofloat
33222Borofloat, http://www.us.schott.com/borofloat/english/attribute/optical,
and the other is a crown glass labeled P-BK7333P-BK7,
http://www.schott.com/advanced_optics/us/abbe_datasheets/schott_datasheet_all_us.pdf,
and the transmission of UV light differs for these two products.

Given the different wavelength dependencies of the above components, our aim
was to measure the four combinations of mirrors and corrector rings present in
the FDs. This meant calibrating at three of the four FD buildings. Table 1
shows the eight telescopes we calibrated at the three FD sites along with
which components make up each telescope. Calibration of these eight telescopes
gives a complete coverage of the different components and a duplicate measure
of each combination.

Table 1: List of the FD telescopes we calibrated and their respective optical components. Calibration at these eight FD telescopes gives a complete coverage of the different components and a duplicate measure of each combination. The last column indicates all other (unmeasured) telescopes with the same optical components. FD telescope | Mirror Type | Corrector Ring | FDs with same
---|---|---|---
|  |  | components
Coihueco 2 | Glass | BK7 |
Coihueco 3 | Glass | BK7 | CO2/3
Coihueco 4 | Glass | Borofloat 33 | CO1,4-6, LA,
Coihueco 5 | Glass | Borofloat 33 | HEAT
Los Morados 4 | Aluminum | Borofloat 33 |
Los Morados 5 | Aluminum | Borofloat 33 | LM
Los Leones 3 | Aluminum | BK7 |
Los Leones 4 | Aluminum | BK7 | LL1-6

As seen in Table 1, the telescopes CO 4/5 are the only ones that have same
nominal components as those located at other FD buildings, which have
different construction dates. It is usualy the case that optical components
degrade their properties when exposed to light and ambient conditions
(ageing), whose effect depends on exposure time. Even if FD telescopes are
kept in climate-controlled buildings, an analysis of ageing follows. Regarding
the spectral calibration, what has to be evaluated is the change in the
spectral response of a given FD telescope, i.e. the shape of the response
curve vs wavelength. This kind of differential degradation is not obviously
seen at the FD telescopes. One way to evaluate whether there is any change in
the spectral response is to track the absolute calibration done periodically
at 375 nm [1, 2]. The absolute calibration is scaled at any given date by
using the nightly relative calibration, which is done at 470 nm [1, 2].
Because these two calibrations are done at different wavelengths, any change
in the spectral response would translate in a drift of the absolute
calibration with time. In Table 2 we show the variations of the ratio (R) of
absolute calibrations performed in 2010 and 2013, where
$R=(Abs_{\mathrm{2013}}-Abs_{\mathrm{2010}})/Abs_{\mathrm{2013}}$, along with
the date of finished construction for telescopes at a given building. As seen
in the table, the variations do not respond to any ageing pattern, e.g. for
the oldest telescopes there is a positive variation for LL and a negative
variation for CO. Moreover, the overall effect that telescopes could have in
data analysis do not change the final reconstructed energy significantly (see
Fig. 49 in [1]). For these reasons, we consider that different time of
telescope construction do not play a role in the spectral calibration
described in this paper and, consequently, CO 4/5 can be taken as
representative of LA and HEAT.

Table 2: List of FD buildings and dates when construction was finished and operation started. $\Delta$t is the elapsed time until measurements done for this work (March 2014). R is the ratio of absolute calibrations performed in 2010 and 2013 (see text). FD building | Built |  $\Delta$t [yr] | R [%]
---|---|---|---
Los Leones | 5/2004 | 9.8 | \+ 1.4
Coihueco | 5/2004 | 9.8 | – 1.6
Los Morados | 3/2005 | 9.0 | – 0.5
Loma Amarilla | 2/2007 | 7.1 | – 0.8
HEAT | 9/2009 | 4.5 |

##  3 Monochromator Drum Setup

The work described in [9] was the first in-situ end-to-end measurement of the
FD efficiency as a function of wavelength. It limited the measurement to only
five points across the $\sim$150 nm wide acceptance of the FDs, and the
filters had a fairly wide spectral width, about $\sim$15 nm FWHM, as shown in
the bottom of Figure 4. The large spectral width led to a complicated
procedure of accounting for the width effects along the rising and falling
edges of the efficiency curve [9]. In addition, since there were only five
measured points, the resulting calibration curve had to be interpolated
between the points, and the original piece-wise efficiency curve [11] was used
as the starting point. In the five-point measurement [9] the efficiency was
assumed to go to zero below 290 nm and above 425 nm since the filters did not
extend to these wavelengths, thus the values resulting from the piece-wise
convolution of the component efficiencies [11] were the only data for
wavelengths below 290 nm and above 425 nm.

Figure 4: A comparison showing the spectral width of the output of the
monochromator sampled every 5 nm (top, this work) and the notch filter
spectral transmission (bottom, [9]). The y-axes are the intensity in arbitrary
units for the monochromator and the normalized transmission for the notch
filters.

These reasons are the motivation for using a monochromator to select the
wavelengths out of a UV spectrum. A monochromator allows for a high resolution
probe across the FD acceptance, and a far more detailed measurement can be
performed. The top of Figure 4 shows the output of the monochromator in 5 nm
steps from 275 nm to 450 nm with a xenon flasher as the input, each step with
a 2 nm FWHM. The xenon flasher is an Excelitas PAX-10 model444PAX-10 10-Watt
Precision-Aligned Pulsed Xenon Light Source -
http://www.excelitas.com/downloads/dts_pax10.pdf with improved EM noise
reduction and variable flash intensity. The monochromator output width was
chosen to provide a reasonable compromise between wavelength resolution and
the drum intensity required for use at the FDs.

For the work described here, an enclosure housing the monochromator and xenon
flasher was mounted onto the rear of the drum. The enclosure was insulated and
contained a heater and associated controlling circuitry to maintain a stable
20$\pm$2 ℃ temperature for monochromator reliability.

A custom 25.4 mm diameter aluminum tube was fabricated and attached to the
output of the monochromator; it protrudes into the interior of the drum. At
the end of the tube a 0.23 mm thick Teflon diffuser ensured that the
illumination of the front face of the drum was uniform as measured with long-
exposure CCD images, similar to what had been measured previously [3, 13].

A photodiode (PD) was mounted near the output of the monochromator, but
upstream of the tube that protruded into the drum, allowing for pulse-by-pulse
monitoring of the emission spectrum from the monochromator. The monochromator
and xenon flasher were controlled with the same common gateway interface (CGI)
web page and calibration electronics that have been used in the absolute
calibration [7]. Scanning of the monochromator, triggering of the flasher, and
data acquisition from monitoring devices and the FD were all fully automated
using CGI code and cURL555cURL Documentation - http://curl.haxx.se/ scripts
over the wireless LAN used for drum calibrations.

##  4 Lab Measurements and the Drum Spectrum

To characterize the drum emission as a function of wavelength, several
measurements were needed in the laboratory. For the one-week calibration
campaign described here, four measurements were performed in the lab, two
prior to any field work at the FD telescopes, one two days later and the last
one at the end of the week.

###  4.1 Drum Emission

With the automated scanning of the monochromator and data acquisition we took
measurements of the relative drum emission spectrum as viewed by the
calibration Lab-PMT. The monitoring PD detector measured the monochromator
output as described above. The setup for these measurements had the drum at
the far end of the dark hall and the Lab-PMT inside the darkbox in the
calibration room, about 16 m away from the Teflon face of the drum. See [7]
for a detailed description of the dark hall calibration setup.

The average response of the Lab-PMT to 100 pulses of the drum was recorded as
a function of wavelength from 250 nm to 450 nm, in steps of 1 nm. The
uncertainty in the average for a given wavelength was calculated as the
standard deviation of the mean, $\frac{\sigma_{\mathrm{Drum}}}{\sqrt{100}}$.
The solid grey line in Figure 5 shows an example of one of these spectra. We
took averages of the four spectra at each wavelength as the final measurement
of the drum spectrum, which is shown in the same figure as blue dots. This
final drum spectrum used measurements in steps of 5 nm corresponding to the
step size used when calibrating the FD telescopes. For wavelengths between 320
nm and 390 nm, the four measurements were generally statistically consistent.
But for wavelengths at the low and high ends of the spectrum there was
disagreement; section 4.3 explains how we introduce a systematic uncertainty
to account for this disagreement.

Figure 5: Drum emission spectra. Solid grey line: one of the measured spectra
taken with the Lab-PMT; the line shows the average responses to 100 pulses of
the drum as a function of wavelength, in steps of 1 nm. Blue points: the
averaged drum spectrum as measured by the Lab-PMT throughout the calibration
campaign; the spectrum is taken in steps of 5 nm as this is what is used to
measure the FD responses; error bars are the statistical uncertainties, which
are generally smaller than the plotted points. Black line and points: the
averaged drum spectrum as measured by the monitoring photodiode (PD)
throughout the calibration campaign, in steps of 5 nm.

For each of these four spectra measured with the Lab-PMT there are data from
the monitoring PD. The monitoring PD data were handled in the same way; we
made an average of the four spectra recorded by the PD and an associated error
based on the spread of the four measurements. These data are shown in Figure 5
as black line and points.

###  4.2 Lab-PMT Quantum Efficiency

A measurement of the quantum efficiency (QE) of the Lab-PMT, which is used to
measure the relative drum emission spectrum, has to be performed to measure
the relative response of a given FD telescope at different wavelengths. The
method used here is similar to what was done previously [9] except, instead of
a DC deuterium lamp, we used the xenon flasher as the UV light source into the
monochromator. For the work reported here we only needed a relative
measurement of the QE, and so several uncertainties associated with an
absolute QE measurement are not included in this work.

Figure 6: Shown in black squares is the measured relative Lab-PMT QE. The
error bars are the statistical uncertainty associated with the distribution of
the response of the PMT at each wavelength. The blue circles are a fourth
order polynomial fit to the data that serves to smooth out the measurement.

Several measurements of the Lab-PMT QE were performed prior to and after the
FD spectral calibration campaign, and these measurements typically yielded
curves consistent with the data shown as black squares in Figure 6. The error
bars are the statistical uncertainty associated with the spread in the
response of the PMT to 100 pulses at each wavelength. The variations in the QE
from point to point are typical when this kind of measurement is performed
(e.g. see [9]), although they are not expected. In an attempt to smooth out
these variations we fit the PMT QE curve with a fourth order polynomial shown
as blue circles in the figure. The error bars in the fit are the relative
statistical uncertainty for a given wavelength applied to the interpolated
values in the fit. Deviations of the fit from measured points are largest at
both the lower and upper ends of the wavelength range. However, the FD
response is significant only in the range 310-410 nm (see Figure 9) where the
deviations are less than 2% with RMS of approximately 1%. We take this 1% as a
conservative estimate of the systematic uncertainty in the measurement of the
Lab-PMT QE: $\delta^{\mathrm{Drum}}_{\mathrm{QESyst}}(\lambda)\approx 1\%$.

Changing the nature of the fitted curve or using simply the measured black
points from Figure 6 has little effect on measurements of EAS events. For
example a change of order 0.1% on the reconstructed energy would result from
using the measured QE points instead of the smoothed curve. The small effect
on energy occurs because in the region at high and low wavelengths where the
fit deviates most from the measured points the FD efficiency is very low and
the nitrogen fluorescence spectrum has no large features.

###  4.3 Uncertainties in Lab Measurements

The estimate of the statistical uncertainties for the various response
distributions to the xenon flasher are taken as the standard deviation of the
mean. Figure 7 shows the response distribution of the Lab-PMT to 100 flashes
of the drum at 375 nm where
$\delta^{\mathrm{Drum}}_{\mathrm{PMTStat}}(\lambda=375\text{
nm})=\frac{\sigma(\lambda=375\text{ nm})}{\sqrt{N}}\approx 1\%$ of the average
response $\overline{S^{\mathrm{Drum}}}(\lambda=375\text{nm})$. The intensity
at the monochromator output is known to be stable (with associated statistical
uncertainties) through the monitoring PD spectra taken at the same time as the
Lab-PMT data. A similar distribution was produced for each wavelength in the
Lab-PMT QE measurement and gives
$\delta^{\mathrm{Drum}}_{\mathrm{QEStat}}(\lambda)\approx 1\%$.

Figure 7: Distribution of the response of the Lab-PMT to 100 flashes of the
drum at 375 nm.

Estimating the systematic uncertainties associated with the relative drum
emission spectrum is done by comparing the different drum emission spectra
measured using the Lab-PMT over the course of the one-week campaign. Prior to
the comparison, the Lab-PMT data are normalized by the simultaneous PD data at
each wavelength to account for changes in the monochromator emission spectrum.
Shown in the top panel of Figure 8 are the four drum spectra measured with the
Lab-PMT that are used to calculate an average spectrum of the drum, and the
middle plot shows the residuals from the average in percent as a function of
wavelength.

Over most of the wavelength region where the FD efficiency is nonzero, 300 nm
to 420 nm, the residuals plotted in Figure 8 are close to agreement with each
other within the statistical uncertainties. To estimate the systematic
uncertainty of the drum emission at each wavelength we introduce an additive
parameter, $\varepsilon^{\mathrm{Drum}}_{\mathrm{Syst}}(\lambda)$, such that
calculating a $\chi^{2}$ per degree of freedom comparison via equation (1)
gives $\chi^{2}_{\mathrm{ndf}}\lesssim 1$, and then this parameter is taken as
the systematic uncertainty:

$\chi^{2}_{\mathrm{ndf}}(\lambda)=\frac{1}{3}\sum_{n=1}^{4}\frac{\big{(}S^{\mathrm{Drum}}(\lambda)_{n}-\overline{S^{\mathrm{Drum}}}(\lambda)\big{)}^{2}}{\big{(}\delta^{\mathrm{Drum}}_{\mathrm{PMTStat}}(\lambda)_{n}\big{)}^{2}+\big{(}\varepsilon^{\mathrm{Drum}}_{\mathrm{Syst}}(\lambda)\big{)}^{2}}\lesssim
1~{}.$ (1)

In equation (1) the Lab-PMT response (or drum emission) at a given wavelength
is $S^{\mathrm{Drum}}(\lambda)$, the associated statistical uncertainty is
$\delta^{\mathrm{Drum}}_{\mathrm{PMTStat}}(\lambda)$, and the average spectrum
is $\overline{S^{\mathrm{Drum}}}(\lambda)$.

Figure 8: Four drum emission spectra as measured by the Lab-PMT (top),
residuals from the average in percent (middle), the resulting systematic
uncertainty $\varepsilon^{\mathrm{Drum}}_{\mathrm{Syst}}(\lambda)$ shown as a
percent of the PMT response at a given wavelength (bottom).

For a few wavelengths $\chi^{2}_{\mathrm{ndf}}(\lambda)<1$ without adding the
systematic term in the denominator, and the corresponding systematic
uncertainty is set to zero. But most wavelengths result in
$\chi^{2}_{\mathrm{ndf}}(\lambda)>1$ without the added term, so we calculate
the systematic uncertainty for those wavelengths. The result of this procedure
is that the non-zero Lab-PMT systematic uncertainties vary from less than 1%
to approximately 3%, and in the important region from 300 nm to 400 nm the
average systematic uncertainty is, conservatively, about 1%, see the bottom
panel of Figure 8.

As a check, the PD spectra were treated with a similar evaluation of a
systematic uncertainty at each wavelength as in equation (1). The
corresponding systematic uncertainty estimates for the PD would all be
approximately 1% or smaller. But there is no need to assess a systematic
uncertainty on the drum intensity due to the PD since the PD data are only
used to normalize the PMT data to reduce the spread in PMT measurements, and
we use the spread in (normalized) PMT data for the systematic uncertainty.

We estimate the overall systematic uncertainty on the intensity of the drum at
each wavelength based on the QE measurement of the Lab-PMT
($\delta^{\mathrm{Drum}}_{\mathrm{QESyst}}(\lambda)~{}\approx 1\%$ ) and the
four measurements of the drum spectrum
($\varepsilon^{\mathrm{Drum}}_{\mathrm{Syst}}(\lambda)~{}\approx 1\%$). Each
of these uncertainties is conservatively about 1% in the main region of the FD
efficiency and nitrogen fluorescence spectrum, so a reasonable estimate of the
overall systematic uncertainty of the drum intensity is found by adding them
in quadrature: 1.4%.

##  5 FD Measurements

During the March 2014 calibration campaign we measured the response of the
eight telescopes, as specified in Table 1, in steps of 5 nm, over the course
of five days. Data from the monitoring PD were also acquired during the FD
measurements to be able to control for changes in the drum spectral emission.
The procedure for measuring the telescope response to the multi-wavelength
drum was to first scan from 255 nm to 445 nm in steps of 10 nm, and then scan
from 250 nm to 450 nm in steps of 10 nm. At each wavelength a series of 100
pulses from the drum was recorded by the FD data acquisition at a rate of 1
Hz. A full telescope response is then an interleaving of these scans. Later in
analysis, wavelengths that result in essentially zero FD efficiency - at low
and high wavelengths in the scan corresponding to the edges of the nitrogen
spectrum - were dropped and set to zero.

In the previous sections we evaluated the systematic uncertainty in the drum
light source intensity as a function of wavelength. The contributions to this
uncertainty are the spread in the four measurements of the drum intensity over
the week of the calibration campaign and the systematic uncertainty in the
quantum efficiency of the PMT used to measure the drum output.

In this section we evaluate the uncertainty in the responses of the telescopes
to the drum by comparing the responses of telescopes with the same optical
components - see Table 1. We do this comparison because we have not measured
every telescope in the observatory, so we have to develop a single calibration
constant for each wavelength for each of the four sets of optical components
in the table. Then we use these calibration constants for all telescopes with
like components (again, see Table 1), including those not measured. Combining
this uncertainty on the FD response, described below, with the drum emission
systematic uncertainty will give the overall systematic uncertainties on the
spectral calibration of the telescopes. As we will see below, of the four
combinations of optical components in Table 1 three will result in systematics
on FD response well below the systematics from the drum emission, but one pair
of telescopes will have significantly different responses to the drum
resulting in a systematic uncertainty larger than that from the drum
intensity.

###  5.1 FD Systematic Uncertainties Evaluated by Comparing Similar Telescopes

We assume that the FDs built with like components - the same mirror and
corrector ring types - should give similar responses, and to test that
assumption we make a comparison between them to derive a meaningful systematic
uncertainty. To that end we perform a $\chi^{2}$ test and introduce parameters
to minimize the $\chi^{2}$ such that $\chi^{2}_{\mathrm{ndf}}\lesssim 1$ for
$ndf=34$, where there are 35 wavelengths used in the comparison. The
parameters introduced are an overall scale factor $\beta$ that is applied to
one of the FD responses, and then $\varepsilon_{\mathrm{FD}}$ which is an
estimate of a systematic uncertainty that would be needed to account for the
difference between the two telescopes. Thus the raw response of one of the FDs
as a function of wavelength is then
$\beta*\big{(}FD_{\mathrm{Resp}}(\lambda)\pm\delta^{\mathrm{FD}}_{\mathrm{Stat}}(\lambda)\pm\varepsilon_{\mathrm{FD}}\big{)}$
in the comparison, where $\delta^{\mathrm{FD}}_{\mathrm{Stat}}(\lambda)$ is
the statistical uncertainty (small) as mentioned at the end of this section.
The scale factor $\beta$ does not represent a systematic uncertainty, it just
accounts for any overall difference in response between the two telescopes.
This is similar to performing a relative calibration analysis as in [14]
between the two telescopes.

The minimization is done in two steps according to equation (2) where the sum
is over the $N_{\lambda}$ measured wavelength points:

$\chi^{2}_{\mathrm{ndf}}=\frac{1}{34}\sum_{\mathrm{n=1}}^{\mathrm{N_{\lambda}}}\frac{\Big{(}FD_{1}(\lambda)_{\mathrm{n}}-\beta*FD_{2}(\lambda)_{\mathrm{n}}\Big{)}^{2}}{\Big{(}\delta^{\mathrm{FD_{1}}}_{\mathrm{Stat}}(\lambda)_{\mathrm{n}}\Big{)}^{2}+\Big{(}\beta*\delta^{\mathrm{FD_{2}}}_{\mathrm{Stat}}(\lambda)_{\mathrm{n}}\Big{)}^{2}+\Big{(}\beta*\varepsilon_{\mathrm{FD}}\Big{)}^{2}}{}~{}.$
(2)

First a minimum in $\chi^{2}$ is found by setting
$\varepsilon_{\mathrm{FD}}=0$ and allowing the scale factor $\beta$ to vary.
Once $\beta$ has been determined, $\varepsilon_{\mathrm{FD}}$ is allowed to
vary until $\chi^{2}_{\mathrm{ndf}}\lesssim 1$. Prior to the minimization the
$FD_{2}(\lambda)$ response data are normalized by the ratio of the monitoring
PD response as measured at $FD_{1}$ and $FD_{2}$ for a given wavelength. This
serves to divide out any change in intensity of the light source as measured
by the PD just downstream of the monochromator, and this normalization does,
as expected, improve the agreement in response for some telescope pairs.

Table 3: $\varepsilon_{\mathrm{FD}}$ and $\beta$ values obtained via equation (2) for the similarly constructed telescopes. The $\varepsilon_{\mathrm{FD}}$ for a given pair of telescopes is given in percentage relative to the averaged response of the pair of telescopes at 375 nm. FDs |  $\varepsilon_{\mathrm{FD}}$ [%] | $\beta$
---|---|---
Coihueco 2/3 | 0.34 | 0.97
Coihueco 4/5 | 0.48 | 1.02
Los Morados 4/5 | 0.14 | 1.01
Los Leones 3/4 | 1.7 | 1.05

The values for $\varepsilon_{\mathrm{FD}}$ and $\beta$ are listed in Table 3
for each pair of telescopes that are constructed with nominally identical
components, and the systematic uncertainties, $\varepsilon_{\mathrm{FD}}$, are
reported as a percentage of the average response of the two telescopes at 375
nm.

Aside from Los Leones telescopes 3 and 4, the $\varepsilon_{\mathrm{FD}}$
values derived through this minimization technique are all less than 0.5 % ,
and the $\beta$ scale factors are all within about 3 % of unity. The values
obtained for Los Leones, although larger than the others, are still small. By
trying to find a reason for this difference we note that telescope 4 was part
of the Engineering Array (EA, [15]) together with telescope 5. However, both
telescopes were rebuilt after the EA operation, particularly the mirrors were
all replaced by new ones after re-setting the design parameters. So, the
discrepancy between LL 3 and 4 is highly probably not caused by any difference
in used materials and, in any case, is included in the uncertainties.

We use the $\varepsilon_{\mathrm{FD}}$ calculated for a given pair of FD
telescopes as a systematic uncertainty across all wavelengths for all
telescopes of the corresponding construction; see Table 1. These systematic
uncertainties are then normalized by the telescope response at 375 nm and are
added in quadrature with the uncertainties associated with the spread in Lab-
PMT measurements of the drum (about 1% in important wavelength range, a
function of wavelength), and the Lab-PMT QE (1%, not wavelength dependent) to
calculate the overall systematic uncertainty on telescopes of each combination
of optical components. An example result is plotted as the red brackets in
Figure 9 for the Los Morados telescopes 4 and 5; for this pair (and like
telescopes) the overall systematic uncertainty on the FD response is
approximately $\sqrt{1^{2}+1^{2}+0.14^{2}}=1.4\%$, and it is dominated by the
uncertainty in the drum intensity. For the Coihueco instruments the overall
systematic uncertainty is about 1.5%. For the telescopes at Los Leones the
uncertainty from the difference in response between the two telescopes is
larger than the drum-related systematic uncertainties, and the overall
systematic uncertainty on all of the Los Leones telescopes is about 2.2%.

The statistics of the data taken with the drum light source at the FD
telescopes also contribute to the uncertainties on the calibration constants.
The typical spread in the average response of the 440 PMTs to the 100 drum
pulses at a given wavelength is 0.4% RMS, which is much smaller than the
systematic uncertainties. Adding the statistical uncertainty in quadrature
with the systematic uncertainties yields the overall uncertainties on the
calibration constants listed in Table 4, which are the main result of this
work.

Table 4: Overall uncertainties on spectral calibration constants for the pairs of telescopes measured and all other (unmeasured) telescopes with the same optical components. FDs | Overall | FDs with same
---|---|---
| uncertainty [%] | components
Coihueco 2/3 | 1.5 | CO2/3
Coihueco 4/5 | 1.5 | CO1,4-6, LA, HEAT
Los Morados 4/5 | 1.5 | LM
Los Leones 3/4 | 2.2 | LL1-6

###  5.2 Photodiode Monitor Data

We performed a comparison between the average dark hall PD spectrum and each
of the spectra measured for the data-taking nights at the FDs to ensure that
the light source was stable and was consistent with what had been measured in
the lab. An overall correction of $1.00\pm 0.01$ night to night was found as
the average ratio of the PD response at the FD to that at the lab to
accommodate any overall variations in intensity or response due to temperature
effects, and then we performed a $\chi^{2}$ comparison for all the measured
wavelengths. For all measuring nights at the FDs the PD spectra agree very
well, the comparison gives a $\chi^{2}_{\mathrm{ndf}}\sim 1$ where $ndf=34$
for each, implying that the spectrum as observed by the PD was the same at all
locations.

##  6 Calculation of the FD Efficiency

We calculate the relative FD efficiency for each telescope by dividing the
measured telescope response to the drum by the measured drum emission
spectrum. The relative drum emission spectrum is measured as described in
section 4 and takes into account the Lab-PMT quantum efficiency over the range
from 250 nm to 450 nm.

The relative efficiency for a given telescope at a given wavelength,
$FD_{\mathrm{eff}}^{\mathrm{Rel}}(\lambda)$, is calculated for each wavelength
from 280 nm to 440 nm in steps of 5 nm:

$FD_{\mathrm{eff}}^{\mathrm{Rel}}(\lambda)=\frac{FD_{\mathrm{Resp}}(\lambda)*{QE^{\mathrm{Lab}}_{\mathrm{PMT}}}(\lambda)}{\overline{S^{\mathrm{Drum}}}(\lambda)}*\frac{1}{FD_{\mathrm{eff}}(\lambda=375~{}\text{nm})}{}~{}.$
(3)

The curves are taken relative to the efficiency of the telescope at 375 nm
since this is what is used in the Pierre Auger Observatory reconstruction
software [16] for all FD calculations. The range in wavelength from 280 nm to
440 nm used for evaluating the FD efficiency is smaller than the range
measured in the lab because below 280 nm and above 440 nm the light level is
near zero intensity for the nitrogen emission spectrum and the FD response is
also very near zero. As an example, Figure 9 shows the relative efficiency for
the Los Morados telescopes 4 and 5 based on this work compared with the
previous measurement [9].

Figure 9: Relative efficiencies for the average of Los Morados telescopes 4
and 5. Top: the five filter curve shown as a dashed blue line [9] and the
monochromator result shown as the solid line. Error bars are statistical
uncertainties, and the red brackets are the systematic uncertainties
calculated as described in section 5. Bottom: difference between the five
filter result and this work. The error bars and brackets are the same as in
the top plot, shown here for clarity.

The uncertainties in the FD efficiencies have statistical and systematic
components associated with the measurement of the relative emission spectrum
of the drum, the Lab-PMT QE, and the FD response to the multi-wavelength drum.
The statistical uncertainties associated with the lab work and the FD
responses are propagated through the calculation of the FD efficiency via
equation (3) as a function of wavelength. All systematic uncertainties
described above associated with the lab work, the Lab-PMT and its QE, the FD
response, and $\varepsilon_{\mathrm{FD}}$ for a given FD telescope, are added
together in quadrature as a function of wavelength.

For much of the wavelength range the new results agree with the older five-
point scan. The disagreement at the shortest and longest wavelengths is
perhaps not surprising since the previous lowest and highest measurements were
at 320 nm and 405 nm, and the efficiency was extrapolated to zero from those
points following the piecewise curve [11]. The efficiency was assumed to go to
zero below 295 nm and above 425 nm.

##  7 Comparison of telescopes with differing optical components

After estimating the systematic uncertainties for each measured FD telescope,
$\varepsilon_{\mathrm{FD}}$, we made a $\chi^{2}$ comparison between the six
combinations of unlike FD optical components listed in Table 1 to determine
whether the unlike components result in any different telescope responses. In
calculating the $\chi^{2}_{\mathrm{ndf}}$ for the differently constructed FD
telescopes we use the ratio of the PD data taken at the corresponding FDs to
normalize the average response of one of the FD types. The PD data from the
two FD data-taking nights are averaged as a function of wavelength and the
ratio of the PD averages from the two types of FDs are applied to the combined
FD response along with the statistical and systematic uncertainties. Using
this normalization serves to divide out any differences in the drum spectrum
between the two measurements of the FD types. An example for calculating the
$\chi^{2}$ between the average of Coihueco 2 and Coihueco 3
($\overline{S_{\mathrm{CO23}}(\lambda)_{\mathrm{n}}}$) and the average of
Coihueco 4 and 5 follows. The uncertainties in equation 4 have obvious labels;
for example $\varepsilon_{\mathrm{FD}}^{\mathrm{CO23}}$ is the systematic
uncertainty for the Coihueco telescopes 2 and 3 from Table 3.

$\begin{split}&\chi^{2}_{\mathrm{ndf}}=\frac{1}{34}\sum_{\mathrm{n=1}}^{\mathrm{N_{\lambda}}}\frac{\Big{(}\overline{S_{\mathrm{CO23}}(\lambda)_{\mathrm{n}}}-PD_{\mathrm{Ratio}}(\lambda)_{\mathrm{n}}*\overline{S_{\mathrm{CO45}}(\lambda)_{\mathrm{n}}}\Big{)}^{2}}{\big{(}\delta^{\mathrm{CO23}}_{\mathrm{Stat}}(\lambda)_{\mathrm{n}}\big{)}^{2}+\big{(}PD_{\mathrm{Ratio}}(\lambda)_{\mathrm{n}}*\delta^{\mathrm{CO45}}_{\mathrm{Stat}}(\lambda)_{\mathrm{n}}\big{)}^{2}+\big{(}\varepsilon_{\mathrm{FD}}^{\mathrm{CO23}}\big{)}^{2}+\big{(}\varepsilon_{\mathrm{FD}}^{\mathrm{CO45}}\big{)}^{2}}~{},\\\
&\\\
&PD_{\mathrm{Ratio}}(\lambda)\equiv\frac{\overline{S^{\mathrm{PD}}_{\mathrm{CO23}}}(\lambda)}{\overline{S^{\mathrm{PD}}_{\mathrm{CO45}}}(\lambda)}~{}.\end{split}$
(4)

The results of the comparisons are shown in Table 5. The telescopes with
different components are all significantly different from each other except
when comparing the average of Los Leones telescopes 3 and 4 to the average of
Los Morados telescopes 4 and 5. In principle this low $\chi^{2}$ could
indicate that all telescopes constructed with components like those at Los
Leones 3 and 4 and those constructed like Los Morados telescopes 4 and 5 have
the same response, and therefore could share the relative calibration
constants that are the goal of this work. However, the two detectors at Los
Leones have a much greater difference in response between them than do the two
telescopes from Los Morados: the systematic uncertainty in Table 3 for Los
Leones telescopes is more than a factor of 10 larger than that for Los Morados
ones. The large systematic uncertainty for the telescopes at Los Leones could
be masking a real difference with those at Los Morados. For this reason we
feel it is reasonable not to combine the Los Leones and Los Morados telescopes
to calculate the final spectral calibration constants.

We conclude that all four sets of FD telescopes listed in Table 1 need
different spectral calibrations, and four sets of calibration constants have
been computed.

Examining the results in Table 5 and Table 1 we note that the largest
$\chi^{2}_{\mathrm{ndf}}$ values in Table 5 are associated with changing
mirrors not changing corrector rings. For example, comparing Coihueco 4/5 with
Los Morados 4/5 changes only the mirror and gives a $\chi^{2}_{\mathrm{ndf}}$
of 55, but comparing Coihueco 4/5 with Coihueco 2/3, which changes only the
corrector ring, yields a $\chi^{2}_{\mathrm{ndf}}$ of 5.6. Changing both
components by comparing Coihueco 2/3 with Los Morados 4/5 gives a
$\chi^{2}_{\mathrm{ndf}}$ of 161, but we note that the telescopes at Los
Morados have a very small systematic uncertainty in Table 3. These examples
have so far left out the Los Leones telescopes. The large systematic
uncertainty derived by comparing the two Los Leones telescopes reduces the
$\chi^{2}_{\mathrm{ndf}}$ values when comparing to other telescopes, but the
idea that the mirrors are the main effect is still present when comparing the
Los Leones telescopes to the others.

Table 5: Comparison of spectral response for FD telescopes with different components. $\chi^{2}_{\mathrm{ndf}}$ values obtained for the sets in Table 1, where $ndf=34$. Comparison | $\mathbf{\chi^{2}_{\mathrm{ndf}}}$
---|---
Coihueco 2/3 vs. Coihueco 4/5 | 2.4
Los Morados 4/5 vs. Los Leones 3/4 | 0.21
Coihueco 4/5 vs Los Morados 4/5 | 57
Coihueco 2/3 vs. Los Leones 3/4 | 10
Coihueco 2/3 vs. Los Morados 4/5 | 144
Los Leones 3/4 vs. Coihueco 4/5 | 6.7

##  8 Effect on Physics Measurables

To evaluate the effect a new calibration has on physics measurables, we
reconstructed a set of events using the new calibration and compare to results
from that same set of events using the prior calibration. When we did this
exercise upon changing from initial piecewise to the five-point calibration,
the reconstructed energies increased about 4% at $10^{18}$ eV, and the
increase lessened slightly to 3.6% at $10^{19}$ eV [12]. The lessening of the
energy increase due the five-point calibration is understood because much of
the change in calibration was at low wavelengths, and the five-point
calibration makes the FDs less efficient at short wavelengths making the
reconstructed energy higher. The higher energy events make more light, and
they can be detected at greater distances than lower energy events. But at
greater distances more of the short wavelength light will be Rayleigh
scattered away, so the lower wavelengths - and the change in calibration there
- do not affect the higher energy events as much when we change to the new
calibration.

When we change from the five-point to the calibration described here, the
reconstructed energies increase on average over all FD telescopes by about 1%,
and that increase is relatively flat in energy. However, this increase is not
the same at all the telescopes. The increase in reconstructed energy is
greatest at Los Leones, about 2.8% at $10^{18}$ eV falling to 2.5% at
$10^{19}$ eV. For Los Morados the reconstructed event energies increase by
about 1.8% without much energy dependence. For all other telescopes the energy
increases, but those increases are less than 0.35% for all energies.

All these changes in the reconstructed energy are important to know to fully
characterize the telescopes. Regarding the associated uncertainties, they are
all significantly smaller than the uncertainties involved in the energy scale
for the FD telescopes (see Table 3 in [1]), particularly the 3.6% from the
Fluorescence yield and the 9.9% from the FD calibration.

##  9 Conclusions

Determining the spectral response of the Pierre Auger Observatory fluorescence
telescopes is essential to the success of the experiment. A method using a
monochromator-based portable light source has been used for eight FD
telescopes with measurements performed every five nanometers from 280 nm to
440 nm. With the calibration of these eight telescopes, the four possible
combinations of different optical components in the FD were covered, thus
assuring the spectral calibration of all FD telescopes at the observatory.

The uncertainty associated with the emission spectrum of the drum light source
used for the calibration was found to be 1.4%, which is an improvement on our
previous 3.5% [9].

For the present work we compared telescopes with nominally the same optical
components, and we find that such pairs have the same spectral response within
a fraction of a percent - as expected - for three out of the four pairs of
like telescopes. But one pair with like components, the oldest telescopes in
the observatory, shows a significant difference in spectral response.

The overall uncertainty in the FD spectral response is 1.5% for 21 of the 27
telescopes. The overall systematic uncertainty for the remaining six
telescopes is 2.2%, and is somewhat larger on account of the larger difference
between the two telescopes measured.

We also compared the differently constructed telescopes. These comparisons
show significantly different efficiencies as a function of wavelength, with
differences mainly in the rising edge of the efficiency curve between 300 nm
and 340 nm. The differences seem to come mostly from the two different mirror
types, and they are reflected in different calibration constants for each of
the four combinations of optical components.

The new calibration constants affect the reconstruction of EAS events, and we
looked at two important quantities. The primary cosmic ray energy increases by
1.8% to 2.8% for half of the telescopes in the observatory, and for the other
half the change in energy is negligible. The position of the maximum in shower
development in the atmosphere, $X_{\mathrm{max}}$, is not changed
significantly by the change in calibration.

## Acknowledgments

The successful installation, commissioning, and operation of the Pierre Auger
Observatory would not have been possible without the strong commitment and
effort from the technical and administrative staff in Malargüe. We are very
grateful to the following agencies and organizations for financial support:

Argentina – Comisión Nacional de Energía Atómica; Agencia Nacional de
Promoción Científica y Tecnológica (ANPCyT); Consejo Nacional de
Investigaciones Científicas y Técnicas (CONICET); Gobierno de la Provincia de
Mendoza; Municipalidad de Malargüe; NDM Holdings and Valle Las Leñas; in
gratitude for their continuing cooperation over land access; Australia – the
Australian Research Council; Brazil – Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq); Financiadora de Estudos e Projetos (FINEP);
Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro (FAPERJ); São Paulo
Research Foundation (FAPESP) Grants No. 2010/07359-6 and No. 1999/05404-3;
Ministério de Ciência e Tecnologia (MCT); Czech Republic – Grant No. MSMT CR
LG15014, LO1305, LM2015038 and CZ.02.1.01/0.0/0.0/16_013/0001402; France –
Centre de Calcul IN2P3/CNRS; Centre National de la Recherche Scientifique
(CNRS); Conseil Régional Ile-de-France; Département Physique Nucléaire et
Corpusculaire (PNC-IN2P3/CNRS); Département Sciences de l’Univers (SDU-
INSU/CNRS); Institut Lagrange de Paris (ILP) Grant No. LABEX ANR-10-LABX-63
within the Investissements d’Avenir Programme Grant No. ANR-11-IDEX-0004-02;
Germany – Bundesministerium für Bildung und Forschung (BMBF); Deutsche
Forschungsgemeinschaft (DFG); Finanzministerium Baden-Württemberg; Helmholtz
Alliance for Astroparticle Physics (HAP); Helmholtz-Gemeinschaft Deutscher
Forschungszentren (HGF); Ministerium für Innovation, Wissenschaft und
Forschung des Landes Nordrhein-Westfalen; Ministerium für Wissenschaft,
Forschung und Kunst des Landes Baden-Württemberg; Italy – Istituto Nazionale
di Fisica Nucleare (INFN); Istituto Nazionale di Astrofisica (INAF); Ministero
dell’Istruzione, dell’Universitá e della Ricerca (MIUR); CETEMPS Center of
Excellence; Ministero degli Affari Esteri (MAE); Mexico – Consejo Nacional de
Ciencia y Tecnología (CONACYT) No. 167733; Universidad Nacional Autónoma de
México (UNAM); PAPIIT DGAPA-UNAM; The Netherlands – Ministerie van Onderwijs,
Cultuur en Wetenschap; Nederlandse Organisatie voor Wetenschappelijk Onderzoek
(NWO); Stichting voor Fundamenteel Onderzoek der Materie (FOM); Poland –
National Centre for Research and Development, Grants No. ERA-NET-ASPERA/01/11
and No. ERA-NET-ASPERA/02/11; National Science Centre, Grants No.
2013/08/M/ST9/00322, No. 2013/08/M/ST9/00728 and No. HARMONIA
5–2013/10/M/ST9/00062, UMO-2016/22/M/ST9/00198; Portugal – Portuguese national
funds and FEDER funds within Programa Operacional Factores de Competitividade
through Fundação para a Ciência e a Tecnologia (COMPETE); Romania – Romanian
Authority for Scientific Research ANCS; CNDI-UEFISCDI partnership projects
Grants No. 20/2012 and No.194/2012 and PN 16 42 01 02; Slovenia – Slovenian
Research Agency; Spain – Comunidad de Madrid; Fondo Europeo de Desarrollo
Regional (FEDER) funds; Ministerio de Economía y Competitividad; Xunta de
Galicia; European Community 7th Framework Program Grant No.
FP7-PEOPLE-2012-IEF-328826; USA – Department of Energy, Contracts No. DE-
AC02-07CH11359, No. DE-FR02-04ER41300, No. DE-FG02-99ER41107 and No. DE-
SC0011689; National Science Foundation, Grant No. 0450696; The Grainger
Foundation; Marie Curie-IRSES/EPLANET; European Particle Physics Latin
American Network; European Union 7th Framework Program, Grant No.
PIRSES-2009-GA-246806; European Union’s Horizon 2020 research and innovation
programme (Grant No. 646623); and UNESCO.

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