11institutetext: 1 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117
Heidelberg, Germany
2 AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, F-91191 Gif-
sur-Yvette, France
3 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1,
D-85748 Garching, Germany
4 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-
Universität München, Scheinerstrasse 1, 81679 München, Germany
5 Carnegie Observatories, Pasadena, CA 91101, USA
6 Infrared Processing and Analysis Center, California Institute of Technology,
Pasadena, CA 91125, USA
7 Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822,
F-69622, Villeurbanne, France
8 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762
Porto, Portugal
9 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP,
Rua das Estrelas, PT4150-762 Porto, Portugal
10 Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-
UB), Martí i Franquès 1, 08028 Barcelona, Spain
11 Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capitá 2-4,
08034 Barcelona, Spain
12 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain
13 Department of Astronomy, University of Geneva, ch. dÉcogia 16, CH-1290
Versoix, Switzerland
14 Department of Mathematics and Physics E. De Giorgi, University of Salento,
Via per Arnesano, CP-I93, I-73100, Lecce, Italy
15 INFN, Sezione di Lecce, Via per Arnesano, CP-193, I-73100, Lecce, Italy
16 Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris,
France
17 Institut d’Astrophysique de Paris, 98bis Boulevard Arago, F-75014, Paris,
France
18 Aix-Marseille Univ, CNRS, CNES, LAM, Marseille, France
19 Observatoire Astronomique de Strasbourg (ObAS), Université de Strasbourg -
CNRS, UMR 7550, Strasbourg, France
20 European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The
Netherlands
21 INAF-Sezione di Lecce, c/o Dipartimento Matematica e Fisica, Via per
Arnesano, I-73100, Lecce, Italy
22 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, UK
23 Aix-Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
24 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via
Piero Gobetti 93/3, I-40129 Bologna, Italy
25 INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, I-35122
Padova, Italy
26 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025 Pino
Torinese (TO), Italy
27 Department of Mathematics and Physics, Roma Tre University, Via della Vasca
Navale 84, I-00146 Rome, Italy
28 INFN-Sezione di Roma Tre, Via della Vasca Navale 84, I-00146, Roma, Italy
29 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131
Napoli, Italy
30 INAF-IASF Milano, Via Alfonso Corti 12, I-20133 Milano, Italy
31 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
32 Port d’Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra
(Barcelona), Spain
33 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, I-00078 Monteporzio
Catone, Italy
34 Department of Physics ”E. Pancini”, University Federico II, Via Cinthia 6,
I-80126, Napoli, Italy
35 INFN section of Naples, Via Cinthia 6, I-80126, Napoli, Italy
36 Dipartimento di Fisica e Astronomia ”Augusto Righi” - Alma Mater Studiorum
Universitá di Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy
37 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125,
Firenze, Italy
38 Centre National d’Etudes Spatiales, Toulouse, France
39 Institut national de physique nucléaire et de physique des particules, 3
rue Michel-Ange, 75794 Paris Cédex 16, France
40 Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
41 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK
42 ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo,
28692 Villanueva de la Cañada, Madrid, Spain
43 European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma,
Italy
44 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix,
Switzerland
45 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa,
Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal
46 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, PT-1749-016 Lisboa, Portugal
47 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405,
Orsay, France
48 INFN-Padova, Via Marzolo 8, I-35131 Padova, Italy
49 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, I-34131
Trieste, Italy
50 Istituto Nazionale di Astrofisica (INAF) - Osservatorio di Astrofisica e
Scienza dello Spazio (OAS), Via Gobetti 93/3, I-40127 Bologna, Italy
51 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46,
I-40126 Bologna, Italy
52 Dipartimento di Fisica ”Aldo Pontremoli”, Universitá degli Studi di Milano,
Via Celoria 16, I-20133 Milano, Italy
53 INAF-Osservatorio Astronomico di Brera, Via Brera 28, I-20122 Milano, Italy
54 INFN-Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
55 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029
Blindern, N-0315 Oslo, Norway
56 Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The
Netherlands
57 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Drive, Pasadena, CA, 91109, USA
58 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
59 Mullard Space Science Laboratory, University College London, Holmbury St
Mary, Dorking, Surrey RH5 6NT, UK
60 Université de Genève, Département de Physique Théorique and Centre for
Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
61 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin
katu 2, 00014 University of Helsinki, Finland
62 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk
4, 7991PD, Dwingeloo, The Netherlands
63 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71,
53121 Bonn, Germany
64 Dipartimento di Fisica e Astronomia “Augusto Righi” - Alma Mater Studiorum
Università di Bologna, via Piero Gobetti 93/2, I-40129 Bologna, Italy
65 INFN-Sezione di Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy
66 Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham, DH1 3LE, UK
67 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France
68 CEA Saclay, DFR/IRFU, Service d’Astrophysique, Bat. 709, 91191 Gif-sur-
Yvette, France
69 Sorbonne Universités, UPMC Univ Paris 6 et CNRS, UMR 7095, Institut
d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France
70 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade
120, DK–8000 Aarhus C, Denmark
71 Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1,
Canada
72 Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada
73 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5,
Canada
74 Institute of Space Science, Bucharest, Ro-077125, Romania
75 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Str. 1, 85741
Garching, Germany
76 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2,
34151 Trieste, Italy
77 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c/o Dipartimento di Fisica,
Edificio G. Marconi, I-00185 Roma, Italy
78 Dipartimento di Fisica e Astronomia “G.Galilei”, Universitá di Padova, Via
Marzolo 8, I-35131 Padova, Italy
79 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain
80 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Tapada da Ajuda, PT-1349-018 Lisboa, Portugal
81 Universidad Politécnica de Cartagena, Departamento de Electrónica y
Tecnología de Computadoras, 30202 Cartagena, Spain
82 INAF-IASF Bologna, Via Piero Gobetti 101, I-40129 Bologna, Italy
83 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700
AV Groningen, The Netherlands
84 Dipartimento di Fisica, Universitá degli Studi di Torino, Via P. Giuria 1,
I-10125 Torino, Italy
85 INFN-Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy
86 Space Science Data Center, Italian Space Agency, via del Politecnico snc,
00133 Roma, Italy
87 INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste TS, Italy
88 SISSA, International School for Advanced Studies, Via Bonomea 265, I-34136
Trieste TS, Italy
89 Departamento de Astrofísica, Universidad de La Laguna, E-38206, La Laguna,
Tenerife, Spain
90 Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, E-38204, San
Cristóbal de La Laguna, Tenerife, Spain
91 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université
de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, F-31400 Toulouse, France
92 Dipartimento di Fisica - Sezione di Astronomia, Universitá di Trieste, Via
Tiepolo 11, I-34131 Trieste, Italy
93 Dipartimento di Fisica e Scienze della Terra, Universitá degli Studi di
Ferrara, Via Giuseppe Saragat 1, I-44122 Ferrara, Italy
94 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, I-40129 Bologna,
Italy
95 INFN-Bologna, Via Irnerio 46, I-40126 Bologna, Italy
96 Department of Physics and Astronomy, University of California, Davis, CA
95616, USA
97 University of Lyon, UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, France
98 INFN-Sezione di Genova, Via Dodecaneso 33, I-16146, Genova, Italy
99 INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del
Cavaliere, 100, I-00100 Roma, Italy
100 Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK
101 School of Physics, HH Wills Physics Laboratory, University of Bristol,
Tyndall Avenue, Bristol, BS8 1TL, UK
102 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, E-28049
Madrid, Spain
103 Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, Carrer de Can
Magrans, s/n Cerdanyola del Vallés, 08193 Barcelona, Spain
104 Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland
105 Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
106 Université PSL, Observatoire de Paris, Sorbonne Université, CNRS, LERMA,
F-75014, Paris, France
107 Code 665, NASA Goddard Space Flight Center, Greenbelt, MD 20771 and SSAI,
Lanham, MD 20770, USA
108 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of
Helsinki, Helsinki, Finland
109 Centre de Calcul de l’IN2P3, 21 avenue Pierre de Coubertin F-69627
Villeurbanne Cedex, France
110 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2,
I-00185 Roma, Italy
111 Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, D-
69120 Heidelberg, Germany
112 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg
16, 69120 Heidelberg, Germany
113 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical
Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780
Bochum, Germany
114 Institute for Computational Science, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland
115 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008,
Santiago, Chile
116 Department of Astrophysical Sciences, Peyton Hall, Princeton University,
Princeton, NJ 08544, USA
117 Department of Physics, P.O.Box 35 (YFL), 40014 University of Jyväskylä,
Finland

#  Euclid preparation. XVIII. The NISP photometric system

Euclid Collaboration: M. Schirmer1<EMAIL_ADDRESS>K. Jahnke1 G. Seidel1 H.
Aussel2 C. Bodendorf3 F. Grupp3,4 F. Hormuth1 S. Wachter5 P.N. Appleton6 R.
Barbier7 J. Brinchmann8,9 J.M. Carrasco10 F.J. Castander11,12 J. Coupon13 F.
De Paolis14,15 A. Franco14,15 K. Ganga16 P. Hudelot17 E. Jullo18 A. Lançon19
A.A. Nucita14,15 S. Paltani13 G. Smadja7 F. Strafella14,15,20 L.M.G.
Venancio21 M. Weiler11,10 A. Amara22 T. Auphan23 N. Auricchio24 A. Balestra25
R. Bender3,4 D. Bonino26 E. Branchini27,28 M. Brescia29 V. Capobianco26 C.
Carbone30 J. Carretero31,32 R. Casas11,12 M. Castellano33 S. Cavuoti34,29,35
A. Cimatti36,37 R. Cledassou38,39 G. Congedo40 C.J. Conselice41 L.
Conversi42,43 Y. Copin7 L. Corcione26 A. Costille18 F. Courbin44 A. Da
Silva45,46 H. Degaudenzi13 M. Douspis47 F. Dubath13 X. Dupac43 S. Dusini48 A.
Ealet7 S. Farrens2 S. Ferriol7 P. Fosalba11,12 M. Frailis49 E. Franceschi24 P.
Franzetti30 M. Fumana30 B. Garilli30 W. Gillard23 B. Gillis40 C. Giocoli50,51
A. Grazian25 L. Guzzo52,53,54 S.V.H. Haugan55 H. Hoekstra56 W. Holmes57 A.
Hornstrup58 M. Kümmel4 S. Kermiche23 A. Kiessling57 M. Kilbinger2 T.
Kitching59 R. Kohley43 M. Kunz60 H. Kurki-Suonio61 R. Laureijs21 S. Ligori26
P.B. Lilje55 I. Lloro62 T. Maciaszek38 E. Maiorano24 O. Mansutti49 O.
Marggraf63 K. Markovic57 F. Marulli64,24,65 R. Massey66 S. Maurogordato67 Y.
Mellier68,17,69 M. Meneghetti24,65 E. Merlin33 G. Meylan44 M. Moresco64,24 L.
Moscardini64,24,65 E. Munari49 R. Nakajima63 R.C. Nichol22 S.M. Niemi21 C.
Padilla31 F. Pasian49 K. Pedersen70 W.J. Percival71,72,73 V. Pettorino2 S.
Pires2 M. Poncet38 L. Popa74 L. Pozzetti24 E. Prieto18 F. Raison3 J. Rhodes57
H.-W. Rix1 M. Roncarelli64,24 E. Rossetti64 R. Saglia75,4 B. Sartoris76,49 R.
Scaramella33,77 P. Schneider63 A. Secroun23 S. Serrano12,11 C. Sirignano78,48
G. Sirri65 L. Stanco48 P. Tallada-Crespí79,32 A.N. Taylor40 H.I. Teplitz6 I.
Tereno45,80 R. Toledo-Moreo81 F. Torradeflot79,32 M. Trifoglio82 E.A.
Valentijn83 L. Valenziano24,65 Y. Wang6 J. Weller3,4 G. Zamorani24 J.
Zoubian23 S. Andreon53 S. Bardelli24 A. Boucaud16 S. Camera84,26,85 R.
Farinelli82 J. Graciá-Carpio3 D. Maino52,30,54 E. Medinaceli24 S. Mei16 N.
Morisset13 G. Polenta86 A. Renzi78,48 E. Romelli49 M. Tenti65 T. Vassallo4 A.
Zacchei49 E. Zucca24 C. Baccigalupi87,88,76,49 A. Balaguera-Antolínez89,90 A.
Biviano76,49 A. Blanchard91 S. Borgani92,76,49,87 E. Bozzo13 C.
Burigana93,94,95 R. Cabanac91 A. Cappi67,24 C.S. Carvalho80 S. Casas2 G.
Castignani64,24 C. Colodro-Conde90 A.R. Cooray96 H.M. Courtois97 M.
Crocce11,12 J.-G. Cuby18 S. Davini98 S. de la Torre18 D. Di Ferdinando93 J.A.
Escartin3 M. Farina99 P.G. Ferreira100 F. Finelli24,93 S. Fotopoulou101 S.
Galeotta49 J. Garcia-Bellido102 E. Gaztanaga11,103 K. George4 G. Gozaliasl104
I.M. Hook105 S. Ilić106,16 V. Kansal2 A. Kashlinsky107 E. Keihanen104 C.C.
Kirkpatrick61 V. Lindholm104,108 G. Mainetti109 R. Maoli110,33 M.
Martinelli102 N. Martinet18 M. Maturi111,112,113 N. Mauri36,65 H.J.
McCracken17,69 R.B. Metcalf64,24 P. Monaco92,76,49,87 G. Morgante24 J.
Nightingale66 L. Patrizii65 A. Peel44 V. Popa74 C. Porciani63 D. Potter114 P.
Reimberg17 G. Riccio29 A.G. Sánchez3,4 D. Sapone115 V. Scottez17 E.
Sefusatti76,49,87 R. Teyssier116 I. Tutusaus12,11 C. Valieri65 J.
Valiviita117,108 M. Viel76,49,87,88 H. Hildebrandt112

(Received; accepted)

Euclid will be the first space mission to survey most of the extragalactic sky
in the $0.95$–$2.02$ $\mathrm{\SIUnitSymbolMicro m}$ range, to a $5\,\sigma$
point-source median depth of $24.4$ AB mag. This unique photometric data set
will find wide use beyond Euclid’s core science. In this paper, we present
accurate computations of the Euclid $Y_{\scriptscriptstyle\rm E}$,
$J_{\scriptscriptstyle\rm E}$, and $H_{\scriptscriptstyle\rm E}$ passbands
used by the Near-Infrared Spectrometer and Photometer (NISP), and the
associated photometric system. We pay particular attention to passband
variations in the field of view, accounting among others for spatially
variable filter transmission, and variations of the angle of incidence on the
filter substrate using optical ray tracing. The response curves’ cut-on and
cut-off wavelengths – and their variation in the field of view – are
determined with $\sim$0.8 nm accuracy, essential for the photometric redshift
accuracy required by Euclid.

After computing the photometric zeropoints in the AB mag system, we present
linear transformations from and to common ground-based near-infrared
photometric systems, for normal stars, red and brown dwarfs, and galaxies
separately. A Python tool to compute accurate magnitudes for arbitrary
passbands and spectral energy distributions is provided. We discuss various
factors from space weathering to material outgassing that may slowly alter
Euclid’s spectral response. At the absolute flux scale, the Euclid in-flight
calibration program connects the NISP photometric system to Hubble Space
Telescope spectrophotometric white dwarf standards; at the relative flux
scale, the chromatic evolution of the response is tracked at the milli-mag
level. In this way, we establish an accurate photometric system that is fully
controlled throughout Euclid’s lifetime.

###### Key Words.:

Instrumentation: photometers, Space vehicles: instruments

##  1 Introduction

Figure 1: Comparison of the NISP spectral response (shaded) with typical
ground-based NIR passbands, in this case VIRCAM@VISTA (Sutherland et al.
2015). Both sets of response curves account for mirrors, filters, and detector
quantum efficiency (QE). The gray line displays the atmospheric transmission
for a precipitable water vapor of $1$ mm and at zenith, taken from the ESO
VISTA instrument description. The ground-based $Z$ and $K$ bands lie outside
the NISP wavelength range, whereas $Y$, $J$, and $H$ cover approximately half
of the corresponding NISP passbands.

The Euclid mission will observe 15 000 deg2 of extragalactic sky (Euclid
Collaboration: Scaramella et al. 2021) from the Sun–Earth Lagrange Point L2.
It will employ weak gravitational lensing and galaxy clustering – including
baryonic acoustic oscillations and redshift space distortions – as
cosmological probes, to determine the expansion history and growth rate of
cosmic structures over the last 10 billion years. The measurements will be
performed in several tomographic redshift bins, covering the time when the
acceleration of the Universe became important (Laureijs et al. 2011; Euclid
Collaboration: Blanchard et al. 2020). In this way, Euclid addresses the
nature and properties of dark energy, dark matter, gravitation, and the
Universe’s initial conditions. The results should be decisive for the validity
of the $\Lambda$CDM concordance model and General Relativity on cosmic scales.

Among many other data, Euclid must determine the near-infrared (NIR)
photometry of at least one billion galaxies to a relative accuracy of better
than 1.5%. This will establish a NIR photometric reference system that will be
in wide use for arguably the next few decades. To achieve these observations
with Euclid’s 1.2 m telescope and within the planned mission duration of six
years, the NISP instrument (Prieto et al. 2012; Maciaszek et al. 2016) carries
a large focal plane array (FPA) of 16 Teledyne HAWAII-2RG detectors. The NISP
wide-field optical system uses filters with a diameter of $130$ mm, the
largest NIR filters flown in a civilian spacecraft to date.

Euclid’s core cosmology science requires the $0.95$–$2.02$
$\mathrm{\SIUnitSymbolMicro m}$ range to be covered in three passbands
($Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$,
$H_{\scriptscriptstyle\rm E}$; see Fig. 1), with rectangular shape, equal
relative spectral width $\Delta\lambda/\lambda$, and without inter-passband
gaps. While overlapping with the common ground-based $Y$, $J$ and $H$
passbands, the Euclid passbands are about twice as wide as they are not
constrained by atmospheric absorption. The passbands’ flanks – i.e. the
transition regions from out-of-band blocking to full in-band transmission –
must be defined by the filters alone. Together with complementary ground-based
photometry, these passbands enable the calculation of the mean photometric
redshifts (photo-$z$) of the tomographic redshift bins with an accuracy of
$0.002\,(1+z)$ (Laureijs et al. 2011; Euclid Collaboration: Ilbert et al.
2021).

The photo-$z$ accuracy required for Euclid implies that the edges of the NISP
passbands must be known to better than $1.0$ nm. In this paper, we show that
the passbands are blueshifted by up to 6 nm when going from the center of the
focal plane toward its corners. This effect is mostly due to variations in the
angle of incidence (AOI) on the filter surface, and to a lesser degree to a
systematic blueshift of the filter’s transmission toward its edges. Using a
careful assessment of the transmission measurements conducted by the filter
coating manufacturer, and accurate ray tracing methods, we determine the
passbands with sub-nm accuracy anywhere in the field of view; this will also
serve all purposes of legacy science.

This paper is organized as follows. In Sect. 2, we introduce the filter
substrates, the various transmission measurements and their limitations. In
particular, we focus on intrinsic, local variations of the filter
transmission. In Sect. 3, we use optical ray tracing to quantify the AOI on
the filter surface, which blueshifts the passband for non-zero AOIs. In Sect.
4, we use these results to compute the effective filter transmission and its
blueshift toward the corners of the focal plane. In Sect. 5, we compute the
spectral response, i.e. the total system transmission including telescope
optics, filters and detectors. We introduce the NISP photometric system in
Sect. 6, together with transformations to common ground-based NIR photometric
systems. We summarize the main results and data products in Sect. 7.

##  2 Filter characteristics and transmission data

###  2.1 Terminology

####  2.1.1 Beam footprint

With ‘beam footprint’ we refer to the intersection of the optical beam on the
filter surface. The beam footprint is asymmetric in shape and has a decentered
obstruction. It’s geometry is important for the transmission calculations.
More details about the footprint can be found in Sect. 3.1.

####  2.1.2 Passband, transmission and response

With ‘passband’ we refer to the $Y_{\scriptscriptstyle\rm E}$,
$J_{\scriptscriptstyle\rm E}$, and $H_{\scriptscriptstyle\rm E}$ wavelength
intervals transmitted by the NISP filters. The filter passbands are
characterized by their wavelength-dependent ‘local transmission’,
$t(\lambda)$, which varies as a function of position on the filter substrate.
The filters ‘effective transmission’, $T(\lambda)$, is obtained by integrating
$t(\lambda)$ over the beam footprint on the filter substrate (Sect. 4). The
‘response’ – i.e. the total system throughput – is computed by multiplying
$T(\lambda)$ with the transmission of other optical elements and the detector
quantum efficiency (QE) (Sect. 5).

The ‘mean peak transmission’ is computed over the wavelength interval where
the transmission exceeds 97% of its maximum value.

Figure 2: Filter geometry and measurement points. The filter has a diameter of
130 mm (black circle). The blue line shows the baffle, blocking light paths
outside without causing vignetting. The 9-point original measurement apertures
are shown by the black rectangles (to scale). Our refined 37-point
measurements are marked by red dots instead of black rectangles, to avoid
cluttering. The gray annulus displays the 90 mm obstructed beam footprint,
i.e. the part of the filter seen by a source at the center of the FPA. The
obstruction is off-centered due to the telescope’s off-axis design. The
response curves made available online are integrated over this central
footprint. The light of a source mapped onto any of the four corners of the
FPA intersects different parts of the filter; the centers of the
correspondingly shifted footprints are shown by the blue squares.

####  2.1.3 Wavelength-related parameters: Cut-on and cut-off

At the ‘50% cut-on wavelength’ (hereafter: cut-on), the near-rectangular
transmission curve has transitioned halfway from out-of-band blocking to the
mean peak transmission. The 50% cut-off is found correspondingly, at the long-
wavelength end of the passband; specifically, we use cubic spline
interpolation to locate the cut-on and cut-off with a precision better than
$0.1$ nm. We refer to the transition regions as the ‘passband flanks’, which
are approximately centered at the cut-on and cut-off. These quantities can be
computed for both the transmission and the response curves.

We calculate the ‘passband width’ as the wavelength interval between the cut-
on and cut-off. Lastly, we define the passband’s ‘central wavelength’ as

$\lambda_{\rm cen}=\frac{\int\lambda\,T(\lambda)\,{\rm d}\lambda}{\int
T(\lambda)\,{\rm d}\lambda}\,.$ (1)

Because of the nearly rectangular transmission curve, the central wavelength
computed in this way is within less than 0.5 nm of the mid-point between the
cut-on and cut-off.

###  2.2 Filter substrate, dielectric coatings, and blocking

The 130 mm-diameter NISP filter substrates were made by Heraeus (Germany) from
their proprietary ‘Suprasil 3001’ type fused silica, and shaped by Winlight
Optics (France). The filters have a center thickness of $11$–$12$ mm,
depending on the specific filter, with the sky-facing side being slightly
convex with a curvature radius of about 10 000 mm.

The filters carry quarterwave stacks of dielectric layers with alternating
high and low refractive index, defining the filter’s passband through
interference. The coatings were performed by Optics Balzers Jena (OBJ,
Germany; now Materion Balzers Optics). Coating stacks of up to 200 alternating
interference layers of SiO2 and Nb2O5 were deposited, to a total stack height
of up to $20$ $\mathrm{\SIUnitSymbolMicro m}$ per filter side, using a plasma
assisted magnetron sputtering process (PARMS). The coatings block photons
outside the nominal passbands very efficiently (Sect. 5.2). The coating layers
defining a single passband flank are deposited on both sides of the filter,
which becomes relevant in Sect. 3.4. A characteristic effect of interference
filters is the blueshift of the passband when the filter is tilted in a
collimated beam against the surface normal vector. This is because the phase
factor of an individual dielectric layer decreases with increasing AOI
(rienstra1998; Smith 2008).

Figure 3: Variation of the 50% cut-on wavelength of the
$Y_{\scriptscriptstyle\rm E}$-band engineering qualification filter.
Measurements were done with a single-beam photometer in 213 apertures $2\times
2$ mm in size (shown to scale), at AOI $\theta=$$$. The cut-on varies by 8 nm
over the unbaffled area, and is a consequence of the coating layers getting
systematically thinner by about 0.4% toward the outer edge. When the gray
annulus moves across this surface (different object positions in the field of
view), the effective transmission changes by $0.5$–$1.0$ nm. Lines are the
same as in Fig. 2.

###  2.3 Measurement strategy with the PE950

The transmission of all filters – one flight model and two flight spares for
each band – was measured with a Perkin Elmer Lambda 950 (PE950). This is a
double beam, double monochromator, intensity ratio recording spectrophotometer
working in the 175–3 300 nm range. In the NIR regime, the PE950 used by OBJ
has a nominal wavelength reproducibility of $0.05$ nm, an absolute wavelength
accuracy of $\leq 0.32$ nm, and is certified periodically by Perkin Elmer
Corp.

Measurements were taken at 1 nm wavelength steps at nine positions (Fig. 2).
They were repeated for two AOIs, $\theta=$$$ and $\theta=$$$, the latter
approximating the largest AOI realized on the filters. This calibrates the
blueshift of the passband for oblique AOIs. Hereafter, we refer to these data
as the ‘9-point data’.

Higher spatial sampling was achieved for the single $Y_{\scriptscriptstyle\rm
E}$-band ‘engineering qualification model’ filter – not used in NISP – using a
single-beam photometer with lower spectral resolution, to determine the cut-on
at over 200 positions (Fig. 3).

The assumption behind this dual approach was that the coating process would be
repeatable, replicating the topography of the cut-on and cut-off ‘surfaces’
for all filters. The sparsely sampled 9-point PE950 data taken over the full
wavelength range would be complemented by the high spatial resolution
measurements of the cut-on surface. This would be sufficient to create
accurate interpolation models, to compute the effective passbands by
integration over the 90 mm obstructed beam footprints (Fig. 2). We show in
Sect. 2.4 that this assumption is invalid.

###  2.4 Non-repeatability of the coating process and wavelength accuracy of
the PE950

The flight model filters passed all acceptance tests and were integrated in
NISP. Our analysis of the local transmission data, however, shows that the
topography of the cut-on and cut-off surfaces varies considerably, even
between flight models and flight spares of the same passband. Consequently,
the assumption of a repeatable coating process is invalid, and the high
spatial resolution mapping data (Fig. 3) is not suitable to improve the
interpolation models. A more detailed analysis of this will be presented
elsewhere, based on physical models of the coating layer stacks.

Furthermore, we show in the following that the PE950 operated at least
partially outside its wavelength specifications summarized at the beginning of
Sect. 2.3.

Figure 4: Wavelength accuracy of the PE950 spectrophotometer. Left panel:
Joint long-term reproducibility of the cut-on and cut-off wavelengths between
the 9-point measurement runs on the same substrates. The RMS is $\sigma_{\rm
wave}^{\rm long}=1.47$ nm. Right panel: Short-term reproducibility within
single 37-point measurement runs, with $\sigma_{\rm wave}^{\rm short}=0.035$
nm.

####  2.4.1 Inconsistencies in the 9-point transmission data

During the first measurement run of the flight spare filters, not all
$\theta=$$$ data were taken for the $Y_{\scriptscriptstyle\rm E}$\- and
$H_{\scriptscriptstyle\rm E}$-band models. A complete set of the 9-point data
including the $\theta=$$$ setting was repeated at a later time. Having the
same positions at $\theta=0^{\circ}$ measured twice, we can check the
wavelength reproducibility and accuracy of the PE950 (Sect. 2.3). To this end,
we compute the normalized wavelength difference
$\Delta\lambda_{50}=(\lambda_{1}-\lambda_{2})/\sqrt{2}$ for each measurement
position, where $\lambda_{1}$ and $\lambda_{2}$ are the cut-ons in the two
measurement runs; the same is done for the cut-off. Assuming normally
distributed errors, the RMS of $\Delta\lambda_{50}$ is then an estimator of
the PE950’s wavelength accuracy.

We do not detect a systematic wavelength offset between the runs. However, we
find that the long-term RMS, $\sigma_{\rm wave}^{\rm long}$, of the
$\Delta\lambda_{50}$ data from the 9-point measurements exceeds the PE950’s
specification; we refer to this RMS as ’long-term’, because many months have
passed between the measurements (see Sect. 2.4.2 for the short-term RMS). In
particular, $\sigma_{\rm wave}^{\rm long}=1.47$ nm (left panel of Fig. 4) is
higher than the specification of $\leq 0.32$ nm. This implies that both the
PE950’s internal wavelength reproducibility of 0.05 nm as well as the
wavelength accuracy were not met for the flight spare runs. Aging effects are
ruled out since the filter coatings are very durable. We do not have repeated
measurements for the flight model filters, and hence assume that the same
uncertainties apply to them.

####  2.4.2 37-point transmission data on flight spare filters

To better understand these inconsistencies, we designed a refined test
protocol for the second set of flight spare filters that were still accessible
at that time. Transmission was measured at 37 points (red dots in Fig. 2),
including the central position of the original 9-point pattern. The remaining
positions were placed in three rings with diameters of 14, 28, and 48 mm,
respectively. Five of the positions were measured repeatedly, to check for
hysteresis, wavelength reproducibility, and systematic drifts.

We compute the short-term RMS of the $\Delta\lambda_{50}$ data, $\sigma_{\rm
wave}^{\rm short}=0.035$ nm, from the repeatedly visited points (right panel
in Fig. 4). This is within the PE950’s nominal wavelength reproducibility of
0.05 nm, for all three flight spares. Meaningful systematic effects are not
found.

Comparing the passband flanks of the common central position between the
37-point and the 9-point measurements, we find $\sigma_{\rm wave}^{\rm
long}=0.52$ nm ($0.16$ nm) for the first (second) run of 9-point data. This is
above (below) the PE950 specification of $\leq 0.32$ nm for wavelength
accuracy.

####  2.4.3 Conclusion about wavelength accuracy

The origin of these inconsistencies, in particular between the original
9-point measurements (Sect. 2.4.1), remains unclear. It could be due to
temporal instabilities and uncalibrated long-term drifts of the PE950, or
tolerances in the preparation and execution of the measurements. While the
37-point data show that the PE950 can deliver data with high internal
consistency, it is not clear whether this applies to the flight model data,
for which only a single measurement epoch is available. Therefore, we assume
that the 9-point flight model data have the same uncertainties as the 9-point
flight spare data.

The spatial interpolation of the local transmission is based on 9 data points;
in these runs, the wavelength has a statistical uncertainty of $\sigma_{\rm
wave}^{\rm long}=1.47$ nm (Sect. 2.4.1). Integration over the interpolating
function (Sect. 4) has an averaging effect on these uncertainties, plausibly
by a factor of $\sqrt{9}$. Here, we are more conservative and set
$\sigma_{50}=0.8$ nm for the cut-on and cut-off wavelengths of the integrated
transmission in Sect. 4. The passband’s central wavelength, when approximated
as the midpoint between the cut-on and cut-off, has an uncertainty of
$\sigma_{\rm cen}=\sigma_{50}\,/\sqrt{2}=0.6$ nm.

Figure 5: Ray tracing of the NISP optical assembly (NI-OA). Shown are the
obstructed beams for three sources located at the center of the focal plane
and at two opposing edges. The NI-OA consists of four lenses; the filter also
acts as a lens, albeit only weakly. The dichroic element is located in the
pupil plane of the telescope, and is not part of NISP. Notice how the beam
paths cover different parts of the filter surface, because – contrary to the
dichroic – the filter is located outside the pupil plane. This leads to
$10$–$23$% of the passband variations in the focal plane due to local
variations of the filter transmission. Notice how the AOI on the filter
changes for the three sources; this is the dominant cause for passband
variations.

##  3 Root causes of the NISP passband variations

Passband variations, i.e. changes of the cut-on and cut-off as a function of
position in the focal plane, must be accurately known for Euclid to meet its
requirements on photo-$z$ accuracy. Two factors contribute to passband
variations:

  1. 1.

increasingly bluer transmission toward the filter edge, in combination with a
moving beam footprint across the filter surface (up to 23% of the total
blueshift for NISP);

  2. 2.

variations of the AOI on the filter surface, in combination with the filter
coating’s effective refractive index, $n_{\rm eff}$ (up to 90% of the total
blueshift).

Euclid’s telescope is based on a Korsch design (Korsch 1977) in an off-axis
configuration (Laureijs et al. 2011). This leads to a number of effects we
need to account for when quantifying the two factors introduced above:

  * •

a decentered central obstruction of the beam by the secondary mirror (Fig. 2);

  * •

a tilted FPA, and slightly tilted NISP optics;

  * •

elliptical beam footprint shape;

  * •

AOI asymmetries on the planar and curved filter surfaces.

The results of our analysis in this Section are used in Sect. 4, where we
compute the effective filter transmission and its variation across the field
of view.

###  3.1 Beam footprint geometry on the filter surface

Consider Fig. 5, displaying the optical paths for three different point
sources at two edges and in the center of the FPA. The corresponding
footprints on the filter surface have an annular shape with outer and inner
diameters of about 91.0 mm and 27.8 mm, respectively (Fig. 6). The off-axis
configuration leads to a slight ellipticity and off-centered central
obstruction. A single footprint covers $\sim$56% of the filter area, and its
location on the filter surface depends on the source position in the field of
view (Figs. 5, 6); the footprint can move away from the filter center by up to
11.9 mm. In combination with the spatially variable filter transmission (Fig.
3), this causes a dependence of the effective passband on the image field.

Depending on the position on the filter, the major axis of the footprint
changes between 89.8 mm and 92.9 mm. The footprint shapes are nearly circular,
with differences from 0.0 to 1.3 mm between the minor and major axes. The
circular obstruction has a diameter of 27.8 mm for the central footprint, and
scales correspondingly for footprints at the corners. The obstruction is
spatially offset in the beam by 3.7 mm (Figs. 2, 3, 6, 7).

###  3.2 Transmission blueshift due to oblique AOI

Like all optical interference filters, the transmitted passband is blueshifted
with increasing AOI, $\theta$, due to the changing phase-difference of
reflected and incident rays. The blueshift also depends on the coating’s
effective refractive index, $n_{\rm eff}(\lambda)$ (see e.g. Amra et al.
2021). In vacuum, and with the small angle approximation applicable here, a
particular passband feature seen at wavelength $\lambda_{0}$ for
$\theta=0^{\circ}$ is blueshifted to

$\lambda=\lambda_{0}\;\sqrt{1-\left(\frac{\sin\theta}{n_{\rm
eff}(\lambda_{0})}\right)^{2}}\;;$ (2)

for details see Smith (2008) and Löfdahl et al. (2011). The maximum AOI
realized on the NISP filters is $\theta=7.5^{\circ}$; polarization splitting
due to polarization-dependent Fresnel reflections is ignored, as it becomes
important for larger angles, only.

Thus, to compute the passband blueshift for a given footprint, we have to
determine two quantities: $n_{\rm eff}$ from measurements and / or the filter
coating design (Sect. 3.3), and the spatial dependence of $\theta(x,y)$, with
$x$ and $y$ specifying the location where the individual rays intersect the
filter’s surface (Sect. 3.4).

Table 1: Effective refractive index. Columns 2 and 3 show the design values of the coating layer stacks for the cut-on and cut-off wavelengths, and columns 4 and 5 the measured values for the flight model filters. | Design values | Measured values
---|---|---
Filter | $n_{\rm eff}^{\rm cut-on}$ | $n_{\rm eff}^{\rm cut-off}$ | $n_{\rm eff}^{\rm cut-on}$ | $n_{\rm eff}^{\rm cut-off}$
$Y_{\scriptscriptstyle\rm E}$ | 1.764 | 1.759 | $1.75\pm 0.07$ | $1.79\pm 0.12$
$J_{\scriptscriptstyle\rm E}$ | 1.769 | 1.776 | $1.66\pm 0.09$ | $1.78\pm 0.10$
$H_{\scriptscriptstyle\rm E}$ | 1.773 | 1.773 | $1.74\pm 0.05$ | $1.86\pm 0.05$

###  3.3 Effective refractive index $n_{\rm eff}$

While $n_{\rm eff}$ is known from the coating design, imperfections in the
coating process cause thickness variations of the layers, and thus modulate
$n_{\rm eff}$. In Table 1 we list the expected and measured values of $n_{\rm
eff}$, obtained using Eq. (2) by comparing the cut-on and cut-off determined
from the OBJ measurements at $\theta=$$$ and $\theta=$$$ (Sect. 2.3).

The 9-point measurements of $n_{\rm eff}$ for the flight model filters show a
RMS of up to 6%. This can be caused by local thickness variations of the
layers, and by the inherent wavelength uncertainty of the PE950 data. The
37-point measurements – having higher internal consistency – show more subtle
variations of $n_{\rm eff}$ across the flight spare substrates, decorrelating
for spatial distances above $20$–$30$ mm. Since we integrate the effective
passband over a $\sim$90 mm beam footprint, the variations in $n_{\rm eff}$
are smoothed accordingly. We thus adopt the mean of all design values listed
in Table 1, $\langle n_{\rm eff}\rangle=1.769$, similar to the mean value
($1.763$) of the measured data. The error made by this simplification – i.e.
constant $n_{\rm eff}$ – is found negligible (see Sect. 4.4).

Figure 6: The mean AOI, $\hat{\theta}$, computed from the AOI on the front and
back filter side, for different footprint positions on the filter surface. The
coordinate squares represent the largest area on the filter within which the
footprint can move: footprints displayed at the grid corners correspond to the
corners of the FPA. The linear obstructions by the three spider arms – holding
the secondary mirror – are too thin to be seen in this representation. Note
that we actually work with a $9\times 9$ grid; for this figure, every 2nd row
and column has been omitted for clarity.

###  3.4 AOI from Zemax optical ray tracing

As stated above, the sky-facing side of the NISP filters is slightly convex
with a curvature radius of $\sim$10 000 mm, and their rear side is flat; thus
the angles of incidence and exitance are different. As mentioned in Sect. 2.2,
a single passband flank is defined by coatings on both surfaces, with more
details undisclosed to us. Since the angles of incidence and exitance of a ray
differ at most by $[-0.32^{\circ};+0.25^{\circ}]$, we simply use their mean
value, $\hat{\theta}$. The error made by this simplification is estimated in
Sect. 4, by repeating the computation for both angles or filter surfaces.

Using Zemax ray tracing, we determine the distribution of $\theta(x,y)$ on
both filter surfaces for a grid of $9\times 9$ positions in the field of view.
For each position, the pupil plane of the optical system is sampled with a
regular grid of $100\times 100$ rays, some of which are vignetted111The
coordinate system used by Zemax for the filter plane must be rotated by $None$
to match the $(x,y)$ filter substrate coordinate system, in which OBJ
performed the transmission measurements. Figures 6 and 7 use the filter
substrate coordinate system..

The resulting values of $\hat{\theta}(x,y)$ are shown in Fig. 6, where every
second footprint from the $9\times 9$ grid is omitted for clarity. In the
computation of the effective transmission (Sect. 4), we use a 2D spline
interpolation for $\hat{\theta}(x,y)$ for each node in the $9\times 9$ grid.
Averaging $\hat{\theta}$ for each footprint, we find
$\langle\hat{\theta}\rangle=0.5^{\circ}$ for the central position; this fairly
large value is caused by a deliberate small tilt of the NISP optics to
optimize the image quality in the off-axis system. For the footprints in the
bottom and top corners, we find $\langle\hat{\theta}\rangle=6.5^{\circ}$ and
$7.1^{\circ}$, respectively. Within a footprint, $\hat{\theta}$ varies by
about $\pm 0.3^{\circ}$; this is displayed in Fig. 7, where we plot
$\Delta\hat{\theta}=\hat{\theta}-\langle\hat{\theta}\rangle$.

The conclusion of this Section is that $n_{\rm eff}$ and $\hat{\theta}(x,y)$
are well understood and accurately known for the computation of the effective
filter transmission, $T(\lambda)$, in Sect. 4.

Figure 7: Same as Fig. 6, after subtracting $\langle\hat{\theta}\rangle$ of
each footprint. This highlights the variation of $\hat{\theta}$ within the
footprints. Note the different color scale compared to Fig. 6.

##  4 Computing the filters’ effective transmission

###  4.1 Numerical integration with blueshift

####  4.1.1 Computational principle

Let $t(\lambda,x,y)$ be the local filter transmission for
$\theta(x,y)=0^{\circ}$, where $x$ and $y$ are the Cartesian filter
coordinates defined in Fig. 2. For an oblique angle $\theta$, we find the
blueshifted transmission, $t^{\prime}$, by looking up the original
transmission, $t$, at the correspondingly redshifted wavelength
$\lambda^{\prime}$,

$t^{\prime}(\lambda,x,y,\theta)=t\left(\lambda^{\prime}(\theta),x,y\right),$
(3)

with

$\lambda^{\prime}(\theta)=\lambda\left[1-\left(\frac{\sin\theta}{n_{\rm
eff}(\lambda)}\right)^{2}\right]^{-1/2}\;.$ (4)

The scaling factor is just the inverse of the one in Eq. (2), used to compute
the blueshift.

Note that in general $t^{\prime}(\lambda)\neq t(\lambda^{\prime})$, as the
transmission at an oblique angle decreases with respect to normal incidence.
However, for small values of $\theta$, and for angle-tuned coating designs,
this effect can be neglected: For NISP we measure a reduction in mean peak
transmission of $0.03$%, $0.17$% and $0.04$% for $Y_{\scriptscriptstyle\rm
E}$, $J_{\scriptscriptstyle\rm E}$, and $H_{\scriptscriptstyle\rm E}$,
respectively, for $\theta=$$$ with respect to $\theta=$$$, and thus Eq. (3)
remains valid.

Now consider a source image anywhere in the focal plane. The corresponding
chief ray intersects the filter substrate at a point $P$. The effective
transmission, $T(\lambda)$, for the source is given by integrating the local
blueshifted transmission, $t(\lambda^{\prime})$, in the filter plane over the
annular footprint $A$ centered on $P$,

$T(\lambda)=\int\displaylimits_{A}t\left(\lambda^{\prime}(\theta),x,y\right)\,{\rm
d}^{2}A\,.$ (5)

To numerically compute this integral, we construct a spline interpolating
function222Using Mathematica’s Interpolation[] routine. to the 9-point data
that sample $t\left(\lambda^{\prime}(\theta),x,y\right)$. The interpolation is
quadratic in the two spatial dimensions and cubic in the spectral dimension.
And we replace $\theta(x,y)$ by a spline interpolation of the mean AOI,
$\hat{\theta}(x,y)$ (see Sect. 3.4).

####  4.1.2 Simplifications to the beam footprint geometry

To facilitate the numeric integration in Eq. (5), we simplify the footprint
geometry. The outer edge of the footprint is nearly circular, with the largest
ellipticity of $91.8\times 93.0$ mm realized for a source in one of the
corners of the FPA. We approximate the edge with a circle, taking the mean of
the minor and major axis as the radius. The circle’s diameter varies between
89.8 mm and 92.4 mm, depending on position on the filter substrate.

Likewise, the decentered obstruction is considered circular. We approximate it
with a fixed diameter of 27.8 mm, and a fixed spatial offset of $y=+3.7$ mm in
the filter coordinate system, independent of footprint location.

These simplifications alter the cut-on and cut-off wavelengths by less than
0.01 nm, and are ignored in our error budget.

####  4.1.3 Simplifications in the transmission integration

In Sect. 3.4, we show that due to the unknown coating design we must use the
mean AOI, $\hat{\theta}(x,y)$, from the curved and planar filter sides. To
estimate the error made by this approximation, we recompute Eq. (5) for the
central footprint position twice, once using $\theta(x,y)$ for the curved
side, and once for the planar side. The resulting wavelength shifts of the
passbands’ cut-on and cut-off are within $[-0.014;0.017]$ nm with respect to
the computation for $\hat{\theta}(x,y)$. Hence this simplification is valid,
and the error small enough to be ignored in the error budget.

Figure 8: Cut-on and cut-off wavelengths in nm as a function of focal plane
position, cold and in vacuum. The blueshift toward the focal plane corners is
evident. The black lines and colored background show the polynomial fits
introduced in Sect. 4.2; the fit parameters are listed in Table 5, and the
joint residuals are shown in Fig. 9. The fits are based on the $9\times 9$
object positions (small dots) for which we inferred the AOI distribution on
the filter surface using Zemax ray tracing. The shaded squares display the 16
NISP detectors, numbered in yellow from 11 to 44. The position of the number
indicates the location of the (1—1) pixel of a detector. The published
response curves (Sect. 7.2) were computed for the central dot marked with the
white square. R_mosaic $(z,y)$ is a physical coordinate system in the focal
plane, to describe – among others – the detector positions.

###  4.2 Passband variations

####  4.2.1 Mathematical model for the total passband shift

To determine the variation of the cut-on and cut-off across the focal plane,
we evaluate $T(\lambda)$ for each node of the $9\times 9$ grid introduced in
Sect. 3.4. The results are displayed in Fig. 8, for cold conditions and in
vacuum (Sect. 4.3), in the physical R_mosaic $(z,y)$ coordinate system in
which the positions of the NISP detectors are accurately known. The
transformation of the filter coordinate system used in Fig. 6 to the R_mosaic
system is

$\displaystyle z_{\rm R\\_mosaic}$ $\displaystyle=-x_{\rm Filter}$ (6)
$\displaystyle y_{\rm R\\_mosaic}$ $\displaystyle=-y_{\rm Filter}\;,$ (7)

i.e. the panels in Fig. 8 are rotated by $None$ with respect to the panels in
Fig. 6.

For accurate and unbiased photo-$z$ measurements, we need a mathematical
prescription that provides the cut-on and cut-off wavelengths for any position
in the focal plane. To this end, we fit 2D 3rd degree polynomials of the
following form to the cut-on and cut-off surfaces in the focal plane for each
filter,

$\lambda_{\rm 50}^{\rm
on/off}(z,y)=a_{0}+\sum_{i=1}^{3}b_{i}\,z^{\;i}+\sum_{i=1}^{3}c_{i}\,y^{\;i}\;.$
(8)

These fits have residuals with an RMS of $\sigma=0.011$ nm, with a maximum
value of $0.06$ nm (Fig. 9). The coefficients are listed in Table 5 in the
appendix. The inclusion of crossterms did not improve the quality of the fits.

The maximum blueshift, $\Delta\lambda_{\rm VAR}$, seen by the FPA increases
about linearly with wavelength, from $2.7$ nm for the
$Y_{\scriptscriptstyle\rm E}$-band cut-on, to $5.8$ nm for the
$H_{\scriptscriptstyle\rm E}$-band cut-off. These values are also given in
Table 3 and shown in Fig. 10.

By setting $\theta=0$ for all rays in the computation, we switch off the
dependence on the AOI and compute the contribution of the intrinsic filter
variations, only. We find that the latter contribute between $10$–$23$% to the
total focal plane variations.

Note that our published NISP response curves (Sect. 7.2) were computed for an
object at the center of the FPA, near the location where the cut-on and cut-
off have their longest wavelengths (Fig. 8).

Figure 9: Joint wavelength residuals of the polynomial fits to the passband
variations, for all filters (their cut-on and cut-off; see also Fig. 8). The
RMS is $\sigma=0.011$ nm; 92% of the residuals are smaller than 0.02 nm.

####  4.2.2 Passband variations for the field corners due to uncertainties in
$n_{\rm eff}$

In Sect. 3.3, we reviewed the uncertainties and variations in $n_{\rm eff}$.
They enter $T(\lambda)$ through Eq. (4), increasing or decreasing the
blueshift in the field corners with respect to the field center. For a
conservative estimate, we recompute $T(\lambda)$ twice, increasing and
decreasing $n_{\rm eff}$ by 5%. We find the following:

At the center of the FPA, where the typical AOI is $\theta=0.5^{\circ}$, the
passbands always shift by 0.002 nm or less; the published response curves
(Sect. 7.2) are computed for this footprint.

At the corners of the FPA with typical values of $\theta=7^{\circ}$, however,
significant shifts occur. In case $n_{\rm eff}$ is larger (smaller) by 5%, the
blueshift $\Delta\lambda_{\rm neff}$ for $Y_{\scriptscriptstyle\rm E}$,
$J_{\scriptscriptstyle\rm E}$, and $H_{\scriptscriptstyle\rm E}$ increases
(decreases) by 0.2, 0.3 and 0.4 nm, respectively. In relative terms, the
difference of the cut-on wavelength between the center of the FPA and its
corners increases (decreases) by $7$–$10$%; same for the cut-off.

Figure 10: NISP passband flanks and variations. The solid red lines display
the local transmission at the filter center (see Fig. 2). The solid blue line
shows the combined vacuum and temperature shifts on the red curve. The thick
black line shows the effective transmission after integrating over the central
beam footprint (white squares in Fig. 8), including the temperature and vacuum
shifts. The colored areas enclosed by dashed lines show the range of
blueshifts seen by the FPA: the left-most dashed line corresponds to the
bluest FPA corners, and the right-most to the reddest passband near – but not
necessarily coincident with – the FPA center. The gray rectangles show the
maximum spread, $\Delta\lambda_{\rm VAR}^{\rm on/off}$, of the cut-on and cut-
off.

####  4.2.3 Passband variations and effect on source photometry

Without correcting for passband variations, the measured flux for an
individual source depends on its position in the field of view, for two
reasons. First, the blueshifted passband selects a different part of the
source’s spectral energy distribution (SED), increasing or decreasing the
measured flux. Second, the cut-off experiences a larger blueshift than the
cut-on when moving from the center of the FPA to a corner. This ‘passband
compression’ reduces the widths of the $Y_{\scriptscriptstyle\rm E}$-,
$J_{\scriptscriptstyle\rm E}$, and $H_{\scriptscriptstyle\rm E}$-bands by 0.7,
0.8 and 1.2 nm, respectively, typically resulting in a SED-dependent flux
reduction.

To estimate the joint amplitude of both effects, we use the catalogs from the
Euclid SC8 (‘science challenge 8’), simulating 100 deg2 of sky with realistic
SEDs as they will be seen by the Euclid Wide Survey (Euclid Collaboration:
Scaramella et al. 2021). We randomly select 1.8 million stars and 1.1 million
redshifted galaxies from the SC8 catalogs (see Sect.6.2.1 for details), and
compute the difference in source photometry for a position at the center of
the FPA and another one in the most blue-shifted corner. The results are shown
in Fig. 15 for galaxies and Fig. 16 for stars. The effect on individual source
photometry is on the order of $1$–$5$ mmag. The differential photometry for
galaxies shows a very low but also broad tail up to 0.6 mag. This tail is
comprised of galaxies with bright emission lines that shift in – or out – of a
passband as the source moves across the field of view. In principle, if such
emission line galaxies were identified in the survey data, they could be used
to improve the knowledge error of the passbands’ cut-on and cut-off as
tabulated in Table 3.

We note that due to the slightly reduced transmission at oblique angle of
incidence, generic flux losses of up to 0.2% can occur in the corners of the
focal plane (Sect. 4.1.1). This effect is systematic and automatically removed
by the illumination correction procedure in the data processing pipeline.

###  4.3 Temperature and vacuum wavelength shifts

The transmission measured at ambient temperature333In order to not mix up
transmission and temperature, we denote temperature by $\tau$ throughout this
paper., $\tau$, and pressure will change in vacuum and at near cryogenic
temperatures. The temperature dependence is caused by the thermal contraction
of the dielectric coating layers, as well as a change in $n_{\rm eff}$ (e.g.
Tan & Arndt 2000, for SiO2).

To quantify these effects, $Y_{\scriptscriptstyle\rm E}$-,
$J_{\scriptscriptstyle\rm E}$-, and $H_{\scriptscriptstyle\rm E}$-band coating
samples were cryocycled in a vacuum chamber at Martin-Luther-University Halle-
Wittenberg (Germany), while simultaneously recording their transmission. We
determine the cut-on and cut-off in the same manner as for the actual filters.
As expected, the passbands redshift slightly when going into vacuum, and
experience a larger blueshift when cold; the combined effect on the passband
flanks is shown in Fig. 10.

Table 2: Fit parameters to compute the blueshift for cooldown, Eq. (9), and the redshift for the transition to vacuum, Eq. (10). | Cold effect | Vacuum effect
---|---|---
Filter | $p_{1}$ | $p_{2}$ | $q_{1}$ | $q_{2}$
$Y_{\scriptscriptstyle\rm E}$ | $0.155$ | $-7.321\times 10^{-4}$ | $0.0376$ | $1.591\times 10^{-4}$
$J_{\scriptscriptstyle\rm E}$ | $5.603$ | $-5.680\times 10^{-3}$ | $0.2243$ | $-1.464\times 10^{-4}$
$H_{\scriptscriptstyle\rm E}$ | $0.707$ | $-1.083\times 10^{-3}$ | $0.0788$ | $-1.443\times 10^{-5}$

####  4.3.1 Temperature dependence

The temperature-related blueshift is normally computed with the Sellmeier
equation (see Fang et al. 2019, for a review). Here, however, since the
transmission was measured at two temperatures only, we linearly fit the
cooldown blueshift as

$\Delta\lambda_{\rm
Temp}(\tau,\lambda)=\frac{\tau_{1}-\tau}{\tau_{1}-\tau_{0}}\,\left(p_{1}+p_{2}\,\frac{\lambda}{1\,{\rm
nm}}\right)\;,$ (9)

with $\tau_{1}=295$ K and $\tau_{0}=120$ K given by the setup, and $p_{1}$ and
$p_{2}$ are the filter-dependent fit coefficients in Table 2.

The expected in-flight filter temperature is $\tau=132$ K. The central
wavelengths of the $Y_{\scriptscriptstyle\rm E}$-, $J_{\scriptscriptstyle\rm
E}$\- and $H_{\scriptscriptstyle\rm E}$-band filters shift by $-0.60$ nm,
$-2.04$ nm, and $-1.13$ nm, respectively. We correct all wavelengths for their
individual temperature dependence, $\Delta\lambda_{\rm Temp}(\tau,\lambda)$.

Note that $\tau=132$ K is the temperature we use for the passbands
characteristics reported in Table 3. It is unlikely that the passbands need to
be corrected for the actual in-flight temperature that will only be known
after launch; the expected temperature bracket for the NISP optics is
$130$–$135$ K. For reference, a change by $+1$ K in filter temperature would
cause a redshift by $0.004$–$0.012$ nm, only. Furthermore, the error we make
by using the linear fit in Eq. (9) instead of the nonlinear Sellmeier equation
is less than $0.1$ nm, and therefore neglected.

####  4.3.2 Vacuum dependence

In analogy to the temperature dependence, we compute a linear fit of the
passband redshift when going into vacuum,

$\Delta\lambda_{\rm Vac}(\lambda)=q_{1}+q_{2}\,\frac{\lambda}{1\,{\rm nm}}\;.$
(10)

The coefficients are collected in Table 2. The corrections are minor,
amounting to $0.21$ nm, $0.023$ nm, and $0.053$ nm at the central wavelengths
of the $Y_{\scriptscriptstyle\rm E}$-, $J_{\scriptscriptstyle\rm E}$-, and
$H_{\scriptscriptstyle\rm E}$-band filters, respectively. Like for the
temperature dependence, we correct all wavelengths for their individual vacuum
dependence.

###  4.4 Summary of the relevant passband uncertainties

In Sect. 3.3, we discussed the uncertainties of $n_{\rm eff}$ and use a
constant $\langle n_{\rm eff}\rangle=1.769$ for all filters. In Sect. 4.2.2,
we estimate corresponding errors of the cut-on and cut-off of $0.2$, $0.3$ and
$0.4$ nm for filters $Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm
E}$, and $H_{\scriptscriptstyle\rm E}$, respectively, for the corners of the
FPA. This error is negligible at the center of the FPA.

In Sect. 2.4.3, we placed a conservative measurement accuracy for the cut-on
and cut-off wavelengths of $0.8$ nm, applicable to all filters. Adding in
quadrature the errors for $n_{\rm eff}$ as given in the previous paragraph, we
obtain increased total uncertainties in the FPA corners of $\sigma_{50}=0.8$,
0.9 and 0.9 nm, respectively, for the three bands. The estimated uncertainty
for the passbands’ central wavelengths is $\sigma_{\rm cen}=0.6$ nm for any
point in the field of view, for all bands. The uncertainties are listed in
Table 3.

All other uncertainties, and errors made due to simplifications, are below 0.1
nm and neglected.

###  4.5 Effect on photometric redshifts

Since photo-$z$ estimates are one of NISP’s principal purposes, we want to
briefly describe the effect of the derived passband knowledge uncertainties on
the photo-$z$ estimates. Obviously, the effect is highly dependent on the
sources’ redshifted SEDs, and on the uncertainties of complementary optical
passbands used. It is beyond this paper to quantify this accurately.
Nonetheless, a simple example illustrates the overall suitability of the
computed NISP passbands for photo-$z$ purposes:

Consider the $0.95$–$1.21$ $\mathrm{\SIUnitSymbolMicro m}$ range of the
$Y_{\scriptscriptstyle\rm E}$-band (Table 3). The characteristic 4000 Å break
falls within the $Y_{\scriptscriptstyle\rm E}$-band for redshifts
$z\in[1.375,2.025]$. If the 4000 Å break was the sole spectral feature
determining the photo-$z$ of a source, then the uncertainty in the central
wavelength, $\sigma_{\rm cen}=0.6$ nm, would introduce a redshift bias of
$\Delta z=\sigma_{\rm cen}/400\,{\rm nm}=0.0015$, compared to the requirement
of $0.002\,(1+z)=0.0048$ for $z=1.375$. Hence, in this example, the knowledge
error of the passband flanks is sufficient. In practice, there are many
sources of bias, which need to be quantified with realistic simulations for a
wide range of redshifted SEDs, folding in complementary ground-based
observations and their uncertainties (e.g. Euclid Collaboration: Ilbert et al.
2021). We remark that the bias $\Delta z$ can be calibrated, as well as the
effects of the passband variations across the focal plane.

##  5 Total system response

Euclid observes simultaneously with NISP and its Visible Instrument (VIS;
Cropper et al. 2012) using a dichroic beam-splitter. To prepare well-defined
input passbands for both instruments, Euclid relies on a finely orchestrated
balance of mirror coatings, the dichroic, filters, the NISP optical assembly
(NI-OA, see Sect. 5.1.2), and detector QE, as illustrated in Fig. 11, and more
quantitatively in Fig. 12. While the NISP filters alone define the cut-on and
cut-off wavelengths, it is only in conjunction with the other elements that
excellent out-of-band blocking is achieved. The total NISP transmission is
therefore given by

$T_{\rm tot}(\lambda)=T_{\rm Tel}(\lambda)\;\,T_{\rm
NI\text{-}OA}(\lambda)\;\,T_{\rm Filter}(\lambda)\;\,T_{\rm QE}(\lambda),$
(11)

where $T_{\rm Tel}(\lambda)$ is the telescope contribution, accounting for the
mirrors and the dichroic. Dependencies of the individual factors $T$ on the
AOI are negligible. We explain Euclid’s chromatic selection function (Figs. 11
and 12) in the following.

###  5.1 Euclid’s chromatic selection function

####  5.1.1 Telescope transmission

The telescope optics common to VIS and NISP consists of – in this order – the
primary mirror M1, the secondary mirror M2, folding mirrors FoM1 and FoM2, the
tertiary mirror M3, and the dichroic element. M1, M2 and M3 are coated with
protective silver and provide a very broad wavelength coverage longward of
$0.34$ $\mathrm{\SIUnitSymbolMicro m}$. After the dichroic and in the VIS
optical path only, folding mirror FoM3 is also coated with protected silver.

At wavelengths of 0.5 $\mathrm{\SIUnitSymbolMicro m}$ and below, VIS and NISP
have considerable QE, which is undesirable for both instruments: At these
wavelengths, galaxy images are increasingly dominated by intrinsic
substructures, adding noise to the VIS weak lensing shape measurement; and for
NISP, unfiltered UV/blue photons would contaminate the NIR photometry.
Therefore, folding mirrors FoM1 and FoM2 have dielectric coatings including
three layers of gold, each. By means of destructive interference and
absorption, these layers block light shortward of $0.42$
$\mathrm{\SIUnitSymbolMicro m}$, reach maximum reflectivity at 0.50
$\mathrm{\SIUnitSymbolMicro m}$, and maintain it beyond $2$
$\mathrm{\SIUnitSymbolMicro m}$.

The dichroic has a complex wavelength selection function (Figs. 11 and 12). It
cuts out a wavelength range from $0.54$–$0.93$ $\mathrm{\SIUnitSymbolMicro m}$
and reflects it to VIS. Photons within $0.93$–$2.15$
$\mathrm{\SIUnitSymbolMicro m}$ are transmitted to NISP. Photons below $0.54$
$\mathrm{\SIUnitSymbolMicro m}$ are also sent to NISP, and are blocked by the
filters and – partially – by the NI-OA (Sect. 5.2). Photons above $2.15$
$\mathrm{\SIUnitSymbolMicro m}$ are mostly reflected to VIS again. This does
not matter for VIS, as the CCDs do not detect photons longward of $\sim$1.1
$\mathrm{\SIUnitSymbolMicro m}$. Note that the dichroic’s reflectance was not
determined nor specified beyond $2.2$ $\mathrm{\SIUnitSymbolMicro m}$. The
NISP filters block out to $2.9$ $\mathrm{\SIUnitSymbolMicro m}$ (Sect. 5.2),
and the detectors have a QE cut-off at $2.3$ $\mathrm{\SIUnitSymbolMicro m}$
(Sect. 5.1.3).

Figure 11: Chromatic selection function of the Euclid optical elements. Note
that the VIS detectors have zero QE for photons with $\lambda>2.15$
$\mathrm{\SIUnitSymbolMicro m}$. The exact behavior of the dichroic above
$2.2$ $\mathrm{\SIUnitSymbolMicro m}$ is unknown: longer wavelengths could
enter NISP and would be blocked by the filters. See Sect. 5.1 and Fig. 12 for
more details.

The combined transmission, $T_{\rm Tel}(\lambda)$, of this arrangement
entering the NISP instrument is shown as the black line in Fig. 12. For more
details, see also Venancio et al. (2020).

Table 3: Summary of the Euclid NISP AB mag photometric system. Wavelengths and their uncertainties are computed from higher precision data and rounded to one digit. The first half of the table lists the global passband characteristics; ‘Width’ refers to the distance between the 50% cut-on and cut-off wavelengths, $\langle T_{\rm peak}\rangle$ to the mean in-band response, and ZP to the photometric zeropoint in the AB system. The second half of the table lists the total uncertainties of the cut-on and cut-off wavelengths, $\sigma_{50}$, and that of the central wavelength, $\sigma_{\rm cen}$. gray!20 | NISP spectral response characteristics
---|---
Filter |  $0.1$% cut-on |  $50$% cut-on |  $50$% cut-off | 0.1% cut-off | $\lambda_{\rm cen}$ | Width | $\langle T_{\rm peak}\rangle$ | ZP
| [nm] | [nm] | [nm] | [nm] | [nm] | [nm] |  | [AB mag]
$Y_{\scriptscriptstyle\rm E}$ | 937.5 | 949.6 | 1212.3 | 1243.2 | 1080.9 | 262.7 | 0.772 | 25.04
$J_{\scriptscriptstyle\rm E}$ | 1151.1 | 1167.6 | 1567.0 | 1595.0 | 1367.3 | 399.4 | 0.790 | 25.26
$H_{\scriptscriptstyle\rm E}$ | 1495.6 | 1521.5 | 2021.4 | 2056.8 | 1771.4 | 499.9 | 0.782 | 25.21
gray!20 | Wavelength uncertainty (center of FPA) | Wavelength uncertainty (corner of FPA) |
Filter | $\sigma_{50}$ | $\sigma_{\rm cen}$ |  | $\sigma_{50}$ | $\sigma_{\rm cen}$ |  |  |
| [nm] | [nm] |  | [nm] | [nm] |  |  |
$Y_{\scriptscriptstyle\rm E}$ | 0.8 | 0.6 |  | 0.8 | 0.6 |  |  |
$J_{\scriptscriptstyle\rm E}$ | 0.8 | 0.6 |  | 0.9 | 0.6 |  |  |
$H_{\scriptscriptstyle\rm E}$ | 0.8 | 0.6 |  | 0.9 | 0.6 |  |  |

####  5.1.2 NISP optical assembly (NI-OA) transmission

NISP carries four lenses, collectively known as the NI-OA (Bodendorf et al.
2019; Grupp et al. 2019); their transmission was not measured. However, we
have the transmission from witness samples that were present in the coating
chamber, used by industry to demonstrate that a minimum transmission
requirement has been exceeded. The joint transmission of these witness samples
is shown as the pink curve in Fig. 12.

Note that this transmission estimate varies between $0.93$–$0.96$ over the
$0.95$–$2.02$ $\mathrm{\SIUnitSymbolMicro m}$ range. It is unclear whether
these variations are also replicated in the flight model optics that may
deviate by several percent. Steep gradients are not expected across this
wavelength range, since only few coating layers were used. The true
transmission of NI-OA will be constrained further in-flight, once observations
of white dwarf (WD) spectrophotometric standard stars are available (Sect.
6.3.2).

The NI-OA blocks photons below $0.58$ $\mathrm{\SIUnitSymbolMicro m}$, apart
from a narrow transmission window between $0.46$–$0.52$
$\mathrm{\SIUnitSymbolMicro m}$ that overlaps with wavelengths transmitted by
the dichroic. The residual optical transmission into NISP is suppressed by the
filters (Sect. 5.2).

####  5.1.3 NISP detector quantum efficiency

The NISP detectors have a sharp QE cut-off at $2.3$
$\mathrm{\SIUnitSymbolMicro m}$, controlled by the Cd concentration in the
HgCdTe alloy (Rogalski 2005). This provides further rejection of long
wavelengths, in addition to the dichroic sending most photons above $2.15$
$\mathrm{\SIUnitSymbolMicro m}$ to VIS.

The QE was measured for each pixel in steps of 50 nm from $0.60$ to $2.55$
$\mathrm{\SIUnitSymbolMicro m}$ (Waczynski et al. 2016). The relative accuracy
– i.e. the knowledge error of the QE curve shape – across this wavelength
range is 1%. The absolute scaling is uncertain to 5%, since the pure QE is
difficult to disentangle from the detector gain (Secroun et al. 2018, and A.
Waczynski, priv. comm.). For the computation of the out-of-band blocking,
only, we linearly extrapolate the QE to zero from $0.60$
$\mathrm{\SIUnitSymbolMicro m}$ to $0.30$ $\mathrm{\SIUnitSymbolMicro m}$.

The mean QE curves of the detectors are very uniform across the $0.95$–$2.2$
$\mathrm{\SIUnitSymbolMicro m}$ range (Bai et al. 2018), and within just a few
percent of each other. Computed for individual detectors at monochromatic
wavelengths, the pixel-to-pixel RMS is $1.2$–$2.4$% globally, and $1.0$%
locally. Their impact on the passband flanks is entirely negligible.

For this paper, we use the mean QE (blue line in Fig. 12) computed from all
pixels of 14 out of the 16 detectors. We exclude one detector because of its
slightly lower QE below 1.0 $\mathrm{\SIUnitSymbolMicro m}$, and another
because of its increasing QE towards 2.0 $\mathrm{\SIUnitSymbolMicro m}$. The
original QE values between $1.65$–$1.75$ $\mathrm{\SIUnitSymbolMicro m}$ are
excluded from all detectors due to a measurement artifact, and replaced by a
local, linear interpolation.

###  5.2 NISP out-of-band blocking and flux contamination

####  5.2.1 Out-of-band blocking

By requirement, the NISP filters must reject light to better than $10^{-3}$
long- and shortward of their passband; the requirement over the VIS range from
$0.54$–$0.93$ $\mathrm{\SIUnitSymbolMicro m}$ is relaxed to $10^{-2}$, since
these photons are directed to VIS already. The blocking of the flight model
filters was measured locally at the center of the substrates, in steps of 2 nm
from $0.3$ to $2.9$ $\mathrm{\SIUnitSymbolMicro m}$.

With a transmission of $10^{-5}$ to $10^{-7}$ below $0.55$
$\mathrm{\SIUnitSymbolMicro m}$, the filters eliminate the residual optical
transmission from the dichroic and the NI-OA. Between $0.55$
$\mathrm{\SIUnitSymbolMicro m}$ and the filters’ cut-on, and longward of the
filters’ cut-off, the blocking is better than $10^{-3}$–$10^{-4}$.

The NISP total out-of-band blocking, including telescope, NI-OA, filters, and
QE, is shown in Fig. 13. Below $0.9$ $\mathrm{\SIUnitSymbolMicro m}$, the
blocking is $10^{-7}$ or better. Above $2.2$ $\mathrm{\SIUnitSymbolMicro m}$,
the blocking is $10^{-5}$, improving to $10^{-7}$ at $2.4$
$\mathrm{\SIUnitSymbolMicro m}$, albeit there are some uncertainties from
extrapolating the telescope transmission to longer wavelengths. Between $0.9$
and $2.2$ $\mathrm{\SIUnitSymbolMicro m}$, the blocking is at least $10^{-4}$.

####  5.2.2 Flux contamination from out-of-band photons

To estimate the contribution from out-of-band photons to the ‘measured flux’ –
i.e. the total flux reaching the detector including out-of-band light –, we
consider sources with a power-law SED $f_{\nu}\propto\nu^{\,\alpha}$ for
frequency $\nu$. For $\alpha=1$ and $\alpha=-3$ we have a blue and red source
with $J_{\scriptscriptstyle\rm E}-H_{\scriptscriptstyle\rm E}$ colors of
$-0.27$ mag and $+0.84$ mag, respectively, in the AB mag system. We also
consider $\alpha=0$, i.e. sources whose AB mag colors are zero.

We then compute the ‘in-band flux’ by integrating $f_{\nu}\,T(\nu)$ within a
passband’s 0.1% cut-on and cut-off wavelengths (Table 3), with $T(\nu)$ as
shown in Fig. 13. The predicted measured flux is obtained by integrating over
the frequency range corresponding to the $0.35$–$2.50$
$\mathrm{\SIUnitSymbolMicro m}$ interval.

Accordingly, the maximum relative contribution from out-of-band flux – outside
the 0.1% cut-on and cut-off – to the total flux in the
$Y_{\scriptscriptstyle\rm E}$\- and $J_{\scriptscriptstyle\rm E}$-bands is
$1.4\times 10^{-3}$ and $2.3\times 10^{-4}$, respectively, for red SEDs with
$\alpha=-3$. For $H_{\scriptscriptstyle\rm E}$-band, we find a maximum
contribution of $1.9\times 10^{-4}$ for the blue SED with $\alpha=1$. Out-of-
band contamination of the NISP photometry is therefore at most at the level of
2 mmag, and more typically $0.2$ mmag, negligible for virtually all practical
purposes. This is not restricted to wavelengths within $0.35$–$2.50$
$\mathrm{\SIUnitSymbolMicro m}$, since there is essentially no sensitivity or
response outside this range.

Figure 12: Transmission of elements in the NISP optical path. The telescope
component (black line) contains all mirrors and the dichroic; the latter
redirects the $0.54$–$0.93$ $\mathrm{\SIUnitSymbolMicro m}$ range to VIS, and
cuts off at $2.15$ $\mathrm{\SIUnitSymbolMicro m}$. The pink line shows the
transmission of the NI-OA. The blue line shows the interpolated mean detector
QE, originally measured at 50 nm intervals. The thin lines show the effective
filter transmission integrated over the beam footprint, and the shaded areas
the total response (accounting for filter, telescope, NI-OA and QE). The
residual optical transmission from the dichroic and the NI-OA between $0.45$
and $0.54\,$\mathrm{\SIUnitSymbolMicro m}$$ is fully suppressed by the
filters’ out-of-band blocking (see Fig. 13). These curves do not include
effects from particulate contamination. Figure 13: Joint NISP out-of-band
blocking, including flight model filters, telescope, NI-OA and QE. The curves
are the same as the shaded curves in Fig. 12, with the difference that the
blocking was measured at the filter center only, in wavelength steps of 2 nm.
We smoothed the data on a 20 nm baseline to reduce the noise at the lowest
transmission levels. The residual optical transmission from the dichroic and
the NI-OA at $0.45$–$0.54$ $\mathrm{\SIUnitSymbolMicro m}$ (Fig. 12) –
shortward of the VIS bandpass – is suppressed by a factor $\geq 4\times
10^{-7}$. Overall, blocking is excellent across the full wavelength range.

###  5.3 In-flight changes of the total transmission

The transmission of the elements presented in this Section will slowly degrade
with time. In practice, Eq. (11) becomes

$T_{\rm tot}(\lambda,t_{\rm m})=T_{\rm Tel}(\lambda)\;\,T_{\rm
NI\text{-}OA}(\lambda)\;\,T_{\rm Filter}(\lambda)\;\,T_{\rm
QE}(\lambda)\;\,T_{\rm Evol}(\lambda,t_{\rm m})\;.$ (12)

Here, $T_{\rm Evol}(\lambda,t_{\rm m})$ absorbs the combined evolution of the
other four factors with progressing mission time, $t_{\rm m}$. At the end of
the mission, six years after launch, we expect an unrecoverable reduction of
5% in the response, i.e. $T_{\rm Evol}(\lambda)\sim 0.95$. Several factors
contribute:

During launch, particulates in the spacecraft and in the rocket fairing will
be redistributed due to the acoustic and mechanical vibrations. This leads to
an increased contamination of Euclid’s primary mirror, and therefore increased
optical scattering.

Once in orbit at L2, Euclid will be subject to space weathering from radiation
damage and dust pitting that lower the system transmission. Radiation damage,
for example, can darken dielectric coatings, an effect that is minimized by
protective coating layers; see e.g. Sheikh et al. (2008) for the Kepler space
telescope, and also Pelizzo et al. (2021).

Proton and electron radiation can degrade the QE of HgCdTe photodiodes such as
in the NISP HAWAII-2RG detectors, as damage displacement in the alloy
generates recombination centers. The expected total ionization dose at L2 for
the NISP detectors at the end of the mission is 2.5 krad; any reduction in QE,
whether chromatic or achromatic, is expected to be minor. We note that QE
degradation due to radiation damage is highly dependent on the photodiode
architecture, see e.g. Sun et al. (2020) and Crouzet et al. (2020), and can
thus be very different for instruments other than NISP.

Surface pitting by dust and meteoroids (Rodmann et al. 2019) as well as by
electrons (Simonetto et al. 2020) increases the micro-roughness of optical
surfaces, and thus the wavelength-dependent scattering. The sky-facing primary
mirror, in particular, will be subject to surface erosion. The level of dust
pitting at L2 is fairly well known (Grün et al. 1985; Gaia Collaboration:
Prusti et al. 2016).

Material outgassing in vacuum (Green 2001; Chiggiato 2020) will contaminate
Euclid’s optical surfaces, mostly with water ice. This alters the total
transmission by wavelength-dependent scattering, interference, and absorption.
Contrary to radiation damage and surface erosion, this process is reversible
by heating the spacecraft, restoring $T_{\rm Evol}(\lambda,t_{\rm m})$ closer
to 1. We will discuss contamination effects in a forthcoming paper.

There are also indirect effects. Electron bombardment lowers the efficiency of
a spacecraft’s multi-layer insulation thermal blankets (Engelhart et al.
2017), resulting in increasing internal temperatures, and thus a – most likely
negligible – redshift of the passbands (Sect. 4.3).

Apart from the latter temperature effect, the cut-on and cut-off wavelengths
are unaffected by these processes. Primarily, the system response – in
practice, $T_{\rm Evol}(\lambda,t_{\rm m})$ – will gradually reduce over time,
which we track with a tight in-flight calibration program (Sects. 6.3.2 and
6.3.3).

##  6 The NISP photometric system

###  6.1 Photometric zeropoints

In general, the number of photo-electrons created per second in a detector in
the frequency interval $[\nu_{\rm min},\nu_{\rm max}]$ is

$n_{\rm e}=A_{\rm eff}\,\int\displaylimits_{\nu_{\rm min}}^{\nu_{\rm
max}}\frac{f_{\nu}(\nu)}{h\nu}\;T_{\rm tot}(\nu)\;{\rm d}\nu\;,$ (13)

where $A_{\rm eff}$ is the effective collecting area, $f_{\nu}(\nu)$ the
source’s spectral flux density in frequency units, $h=6.602\times 10^{-27}$
erg Hz-1 Planck’s constant, $\nu$ the photon frequency, and $T_{\rm tot}(\nu)$
the total ‘transmission’ – i.e. the probability that a photon entering the
telescope is converted to a photo-electron – as given in Sects. 4 and 5. We
assume a perfect detection and extraction chain, i.e. a photo-electron created
in a pixel is registered by the readout electronics and caught by software
processing.

Euclid’s photometric measurements will be given in the AB magnitude system
(Oke & Gunn 1983), where the relation between monochromatic – i.e. per
frequency interval – AB magnitude and spectral flux density is444We use units
of $1\,{\rm Jy}=10^{-23}$ erg s-1 cm-2 Hz-1 for the spectral flux density, to
avoid cluttering by units in the logarithms.

$m_{\rm AB}=-2.5\;\operatorname{log_{10}}\left(\frac{f_{\nu}(\nu)}{1\,{\rm
Jy}}\right)+8.90\;.$ (14)

This equation is also valid for a broad-band observation in case of a
frequency-flat source spectrum.

We define the photometric zeropoint, ZP, of a Euclid broad-band observation as
the magnitude of a source creating a single photo-electron per second, i.e.
$n_{\rm e}=1$ s-1. Assuming a frequency-flat source spectrum,
$f_{\nu}(\nu)={\rm const}$, we solve Eq. (13) for $f_{\nu}$, replace it in Eq.
(14), and find the ZP as

${\rm ZP}=m_{\rm AB}=+2.5\,\operatorname{log_{10}}\left(\frac{A_{\rm
eff}}{n_{\rm e}\,h}\,\left(1\,{\rm Jy}\right)\,\int\displaylimits_{\nu_{\rm
min}}^{\nu_{\rm max}}\frac{T_{\rm tot}(\nu)}{\nu}\,{\rm d}\nu\right)+8.90\;.$
(15)

Using $A_{\rm eff}=9.926\times 10^{3}$ cm2 (L. Venancio, priv. comm.) for
Euclid,

${\rm ZP}=26.84+2.5\,{\rm log}_{10}\,\int\displaylimits_{\nu_{\rm
min}}^{\nu_{\rm max}}\frac{T_{\rm tot}(\nu)}{\nu}\,{\rm d}\nu\;.$ (16)

Setting $\nu_{\rm min}$ and $\nu_{\rm max}$ to the frequency interval where
$T_{\rm tot}(\nu)$ exceeds $0.1\%$ of the mean peak total transmission (see
Table 3), we derive the following AB mag photometric zeropoints,

$\displaystyle{\rm ZP}_{Y_{\scriptscriptstyle\rm E}}=25.04\pm 0.05\,{\rm
mag},$ (17) $\displaystyle{\rm ZP}_{J_{\scriptscriptstyle\rm E}}=25.26\pm
0.05\,{\rm mag},$ (18) $\displaystyle{\rm ZP}_{H_{\scriptscriptstyle\rm
E}}=25.21\pm 0.05\,{\rm mag}.$ (19)

When computed for the 50% cut-on and cut-off wavelengths, these ZPs decrease
by 0.01 mag.

The uncertainty of these ZPs is dominated by the $5\%$ uncertainty of the
absolute QE measurement (Sect. 5.1.3). For comparison, the absolute accuracy
of the spectrophotometers used to measure the mirror reflectances, and the
NISP filter and dichroic transmissions, is $0.02$–$0.2$%. An initial in-flight
flux scaling will tie the NISP photometric system to a WD spectrophotometric
standard star with known absolute flux (Sect. 6.3.2), constraining the ZPs
much more accurately. Yet, throughout the mission, the ZPs will slowly evolve
as explained in Sect. 5.3.

###  6.2 Transformations to other NIR systems

The NISP photometric system, summarized in Table 3, is geared towards
photo-$z$ measurements. It deviates considerably from common ground-based NIR
systems, such as 2MASS (Carpenter 2001), Mauna Kea Observatories (MKO; Leggett
et al. 2006), VISTA (González-Fernández et al. 2017), and UKIRT (Hodgkin et
al. 2009), the latter being built upon MKO (see also Hewett et al. 2006). All
of these are primarily dictated by the atmospheric transmission windows (Fig.
1).

####  6.2.1 Linear transformations for stars and galaxies

In Appendices B to E we provide linear transformations between these external
photometric systems and the NISP photometric system. The transformations are
based on the Euclid SC8 catalogs, containing realistic SEDs for 100 deg2 of
sky as it will be seen by the Euclid Wide Survey (Euclid Collaboration:
Scaramella et al. 2021). We randomly selected 1.8 million stars and 1.1
million redshifted galaxies from the SC8 catalogs, and computed their colors
in the NISP, 2MASS, MKO, and VISTA photometric systems.

The stellar catalog is comprised of Besançon models at the faint end, and the
observed stars of Pickles & Depagne (2010) – their Table 15 in the online
edition – for the bright end (1.8 million in total). Late M-type stars, and L
and T dwarfs (hereafter ‘MLT types’) were taken from Euclid Collaboration:
Barnett et al. (2019) (61 000 in total).

The galaxy catalog in SC8 is built on the Euclid flagship simulation555
https://www.euclid-ec.org/?page_id=4133, supported by CosmoHub (Tallada et al.
2020). The templates are based on Ilbert et al. (2009), in turn relying on the
templates by Bruzual & Charlot (2003) and Polletta et al. (2007). Included are
effects from two different internal extinction laws (Prevot et al. 1984;
Calzetti et al. 2000).

We then fit – for the galaxies and stars separately – linear relations
including a constant offset, of the form

$Y_{\rm A}=J_{\rm B}+c_{0}+c_{1}\,(Y_{\rm B}-J_{\rm B})\,.$ (20)

In this example, an instrument A $Y$-band magnitude is estimated from
instrument B $Y$\- and $J$-band magnitudes. Simple $\chi^{2}$ minimization
yields the $c_{0}$ and $c_{1}$ coefficients including a 3 $\sigma$ outlier
rejection, and we choose the neighboring bands producing the best fit for each
case.

Higher-order polynomial fits do not significantly reduce the fit residuals. We
also find that when a constant offset $c_{0}$ is included in the fit as we do
here, then there is no significant benefit in using more than two bands. For
the majority – but not all – of the conversions even a two-band linear fit
omitting the offset produces results with errors almost as low as the ones
listed in Appendices E and B.

The fits are performed for the following object classes:

Galaxies: Equations (21) through (33) convert 2MASS, MKO and VISTA NIR
magnitudes to NISP magnitudes. Equations (34) through (42) convert in the
other direction, from NISP magnitudes to 2MASS, MKO and VISTA magnitudes.

Stars without MLT types: Likewise, see Eqs. (43) through (55), and Eqs. (56)
through (64).

MLT types: Likewise, see Eqs. (65) through (77), and Eqs. (78) through (86).

Stars of all spectral types: Likewise, see Eqs. (87) through (99), and Eqs.
(100) through (108).

####  6.2.2 Limitations of the linear transformations

The color range for which the transformations are applicable is given by the
x-axes of the plots in Appendices B to E. The plots encompass 99% of the
sources in the SC8 catalogs, and therefore the validity range can be
considered universal for these simple parameterizations.

The transformations for galaxies have residuals of $0.04$–$0.06$ mag, and are
therefore usable for many purposes needing a simple translation between the
different photometric systems. Notable exceptions are the computed $Z$\- and
$K_{\rm s}$-band magnitudes, where residuals amount to $0.1$–$0.2$ mag.

The transformations for stars of spectral types O to K have residuals of
$0.02$–$0.04$ mag. Notable exceptions are the computed $Z$-band magnitudes,
with residuals as large as 0.11 mag.

The transformations for MLT types have residuals of $0.04$–$0.12$ mag. Notable
exceptions are the $Z$-band magnitudes, with residuals up to 0.38 mag. When
working with such red types, the photometry package (Sect. 6.2.3) should be
used.

The transformations for stars of all spectral types (O to K, MLT) have
residuals of $0.03$–$0.08$ mag. Again, notable exceptions are the $Z$-band
magnitudes with residuals up to 0.25 mag.

As a caveat, we emphasize that the SC8 catalogs are representative for the
Euclid Wide Survey area at galactic latitude $|b|\gtrsim 30^{\circ}$. The
stellar disk population is therefore underrepresented in these catalogs, as
are lines of sight of appreciable dust extinction (Euclid Collaboration:
Scaramella et al. 2021). Our transformations must therefore be used with
caution when working at low galactic latitudes. While the Euclid Wide Survey
does not extend into the galactic plane, observations during e.g. a future
mission extension could do so.

We emphasize that all transformations between filter systems given in this
paper are computed in the AB mag system. The 2MASS magnitudes in Cutri et al.
(2003) are given in the Vega system. Before they can be used in any of our
transformations, they must be converted to AB magnitudes, using e.g. the
conversions by Pons et al. (2019).

####  6.2.3 A Python transformation tool

The linear transformations provided in this paper should be useful for many
general purposes. However, individual objects may deviate by 0.1 mag or more,
in particular for galaxies and very blue or red sources (see Sect. 6.2.2, and
Figs. 24 to 18).

For accurate photometry and filter systems other than the ones presented here,
we recommend to use the external, stand-alone Python
photometry666https://github.com/haussel/photometry package. The NISP response
curves of this paper have been integrated in photometry, which includes a
notebook777See notebooks/Euclid_TU_checks.ipynb in the photometry source tree.
showing how to recompute magnitudes in other passbands for arbitrary stellar
and non-stellar SEDs.

Figure 14: Approximate outline of the Euclid self-calibration field (red
circle) near the North Ecliptic Pole (NEP), which will be observed with an
irregular dither pattern. The star symbol sizes scale with the brightest AB
magnitude in the ATLAS-Refcat2 $riz$ bands (blue; Tonry et al. 2018) and the
2MASS $JHK_{\rm s}$ bands (red, converted to AB mag; Cutri et al. 2003). Cyan
and orange stars show hot (blue) WD spectrophotometric standards, and red
extragalactic spectrophotometric standards, respectively. The large blue
circle indicates the boundary of the Euclid Deep Field North.

###  6.3 In-flight lifetime of the NISP photometric system

####  6.3.1 Validity

At the time of this writing, Euclid is still one year from launch. The
transmission measurements presented here are not expected to change
significantly until launch. The space-grade optical coatings are very durable,
and all hardware is kept in clean conditions in a controlled environment up to
and including launch. Only a small degree of particulate contamination is
expected during this time, which is well under control, and whose additional
contribution to the flux loss will be much smaller than the uncertainty of 5%
for the absolute detector QE.

Therefore, the cut-on and cut-off wavelengths presented in this paper are not
expected to change, neither due to the remaining time on-ground, nor during
flight. What will change over time is – to first order – the overall response
as encoded in the ZPs of Sect. 6.1. This will be tracked using a tight
calibration program outlined in Sects. 6.3.2 and 6.3.3. Hence, the AB
magnitudes of sources reported in future Euclid data releases are unaffected
by – i.e. corrected for – any response losses.

Second order effects arise due to wavelength-dependent changes of the
response. For example, scattering by dust and water ice on optical surfaces is
larger at shorter wavelengths, making objects appear redder than they are. In
Eq. (12), we have absorbed the time dependence of the response in the
additional term $T_{\rm Evol}(\lambda,t_{\rm m})$. The Euclid calibration
program (Sects. 6.3.2 and 6.3.3) has been specifically developed to determine
and monitor $T_{\rm Evol}(\lambda,t_{\rm m})$ with great accuracy. This will
eventually even enable SED-dependent corrections of source fluxes, once our
understanding of the system and $T_{\rm Evol}(\lambda,t_{\rm m})$ has matured
at later data releases.

Conclusively, we expect that the NISP photometric system presented in this
paper will maintain its validity throughout the mission. This should also hold
for the linear transformation equations in Sect. 6.2, whose uncertainties are
dominated by the large diversity of the redshifted source SEDs. In the
following two Sections, we summarize the in-flight maintenance.

####  6.3.2 Absolute calibration observations

For a number of science applications the absolute photometry of NISP
observations must be known to within 5% of established reference photometric
systems. To this end, Euclid will observe early one of several suitable WD
spectrophotometric standard stars on a $5\times 5$ grid per detector and per
filter, before the beginning of the routine science observations; hot, stable
WDs are excellent flux calibrators (Bohlin 2014; Bohlin et al. 2020). Euclid
will also take spectra of a WD on five positions per detector. Furthermore,
one of the Euclid WDs is located in Euclid’s self-calibration field (Sect.
6.3.3), which is visited regularly on a monthly basis.

Our approved Hubble Space Telescope (HST) program (ID #16702) has begun
obtaining NIR spectra of a total of six WDs in Euclid’s survey area. These
data will tightly connect the NISP photometric system to the well-known HST
photometric system (Bohlin & Deustua 2019). Other spectrophotometric standards
that happen to be in the survey area will be used for additional spot checks
of the absolute calibration.

####  6.3.3 Relative calibration observations

Euclid’s relative photometric accuracy must be better than 1.5%. To fulfill
this requirement – independently of the WD observations – Euclid will observe
a self-calibration field every $25$–$35$ days (Euclid Collaboration:
Scaramella et al. 2021). Figure 14 shows the location of the ‘Self-cal field’
near the North Ecliptic Pole (NEP), within Euclid’s continuous viewing zone.
Each visit will reach a depth of about $25.0$ AB mag, one magnitude deeper
than the Euclid Wide Survey. At each dither point, VIS and NISP images in all
bands will be taken, including a NISP spectroscopic exposure. Over time, the
Self-cal field will become the deepest Euclid field.

The Self-cal field contains about $40\,000$ unsaturated stars in the
$J=16.5$–$24.0$ AB mag interval, with a signal-to-noise ratio of $S/N\gtrsim
10$. Gaia Collaboration: Eyer et al. (2019) have shown that about 9% of all
Gaia stars are variable down to a precision of $5$–$10$ mmag. Even if half of
all stars detected by Euclid were variable, the statistical basis for ZP
monitoring is excellent, also because variability between stars is
uncorrelated. We expect to detect ZP changes down to $1$–$2$ mmag, per NISP
detector, and in two disjoint color bins (SED bins) per filter. In case
needed, we can fold in twice as many galaxies to improve the statistics.

Should the monthly sampling turn out to be insufficient, e.g. in the presence
of faster ZP variations due to material outgassing, then we can include
photometry from sources appearing in the overlap areas of adjacent survey
fields. Euclid observes about 19 adjacent fields every 24 hours.

The monthly visits of the Self-cal field also provide spectroscopic
observations of the WD standard, and of about 2000 field stars with
$S/N\gtrsim 10$ per spectral resolution element. Using statistical stacking of
all spectra, we can detect chromatic changes per spectral resolution element
on the order of 1 mmag, globally for the full focal plane – although not per
detector as with the imaging observations.

Note that these observations can only weakly distinguish which optical surface
is losing transmission, or whether detectors degrade in QE, and due to which
effects. Nonetheless, these calibrations put Euclid in an excellent position
to determine the net effect on the data, as encoded in $T_{\rm
Evol}(\lambda,t_{\rm m})$, maintaining the NISP photometric system throughout
Euclid’s lifetime.

##  7 Summary and data products

###  7.1 Context and main results

Euclid will observe 15 000 deg2 of the darkest extragalactic sky to a median
depth of $24.4$ AB mag for $5\,\sigma$ point sources (Euclid Collaboration:
Scaramella et al. 2021), establishing a photometric reference data set with
considerable legacy value. Using detailed filter transmission measurements and
ray tracing of the optical system, we determine the edges of the spectral
response curves for any position in the focal plane with a – conservatively
estimated – accuracy of 0.8 nm. The out-of-band blocking of the
$Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$ and
$H_{\scriptscriptstyle\rm E}$ passbands is excellent over the $0.3$–$3.0$
$\mathrm{\SIUnitSymbolMicro m}$ range, with out-of-band contributions of $0.2$
mmag for typical power-law SEDs. The main passband properties, and the
polynomial coefficients to compute the passband edges as a function of field
position, are given in Tables 3 and 5, respectively.

We also derived the photometric zeropoints in the AB mag system. The NISP
passbands are about twice as wide as their ground-based counterparts that are
constrained by the atmospheric transmission windows. We provide linear
transformations for stars and galaxies separately, to convert between the NISP
and the ground-based photometric systems. A Python module is available for the
computation of arbitrary transformations and magnitudes.

We strongly recommend authors to use the ‘E’ subscript – as in
$Y_{\scriptscriptstyle\rm E}$-band 888Euclid papers use the following type-
setting in LaTeX, e.g. for $Y_{\scriptscriptstyle\rm E}$:
$Y_{\scriptscriptstyle\rm E}$ – when referring to Euclid passbands and
magnitudes, to avoid any confusion with the ground-based $YJH$ passbands that
have only half the spectral width.

Overall, we designed and built a well-defined photometric system for NISP with
great legacy value. Our rigorous in-flight calibration program puts Euclid in
an excellent position to maintain this photometric system throughout the
mission.

###  7.2 Published data products and versioning

The $Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$ and
$H_{\scriptscriptstyle\rm E}$ spectral responses at the center of the NISP FPA
are available at an ESA
server999https://euclid.esac.esa.int/msp/refdata/nisp/NISP-PHOTO-PASSBANDS-V1.
Included are the transmission for the filters, telescope, and NI-OA, as well
as the mean detector QE, from which the total spectral response was computed;
an excerpt is reproduced in Table 4. For convenience, the tables are available
in both ASCII and FITS (Hanisch et al. 2001) format, and can be readily
displayed with e.g. TOPCAT (Taylor 2005) and processed with astropy (Astropy
Collaboration: Robitaille et al. 2013; Astropy Collaboration: Price-Whelan et
al. 2018).

We registered an Digital Object Identifier
(DOI)101010https://doi.org/10.5270/esa-kx8w57c for the electronic response
tables (version v1.0) published in this paper. In case updated response curves
become available, a new DOI will be registered for them. The landing web page
for the old DOI will inform the visitor that a new version is available. Older
versions will be retained for reference.

Table 4: Excerpt of the published NISP response and transmission curves (abridged, reduced numeric precision). The columns give the wavelength in nm, total system response, filter transmission, telescope transmission (mirrors and dichroic), transmission of the NI-OA, and detector QE. The response in column 2 is the product of columns 3 to 6. WAVE | T_TOT | T_FILT | T_TEL | T_NIOA | T_QE
---|---|---|---|---|---
943 | 0.0276 | 0.0372 | 0.8670 | 0.9383 | 0.9121
944 | 0.0442 | 0.0594 | 0.8680 | 0.9393 | 0.9126
945 | 0.0619 | 0.0831 | 0.8676 | 0.9403 | 0.9132
946 | 0.0834 | 0.1117 | 0.8686 | 0.9412 | 0.9137
947 | 0.1244 | 0.1660 | 0.8698 | 0.9422 | 0.9142
948 | 0.1913 | 0.2548 | 0.8704 | 0.9431 | 0.9147
949 | 0.2969 | 0.3944 | 0.8716 | 0.9440 | 0.9151
950 | 0.4310 | 0.5714 | 0.8719 | 0.9449 | 0.9156
951 | 0.5617 | 0.7425 | 0.8730 | 0.9457 | 0.9162
952 | 0.6626 | 0.8739 | 0.8739 | 0.9465 | 0.9167
953 | 0.7178 | 0.9443 | 0.8748 | 0.9473 | 0.9173
954 | 0.7362 | 0.9658 | 0.8760 | 0.9481 | 0.9178
955 | 0.7411 | 0.9706 | 0.8763 | 0.9488 | 0.9183
956 | 0.7425 | 0.9705 | 0.8768 | 0.9496 | 0.9189
957 | 0.7438 | 0.9702 | 0.8776 | 0.9503 | 0.9194

###### Acknowledgements.

The Euclid Consortium acknowledges the European Space Agency and a number of
agencies and institutes that have supported the development of Euclid, in
particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian
Science Policy, the Canadian Euclid Consortium, the French Centre National
d’Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish
Space Research Institute, the Fundação para a Ciência e a Tecnologia, the
Ministerio de Economia y Competitividad, the National Aeronautics and Space
Administration, the National Astronomical Observatory of Japan, the
Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the
Romanian Space Agency, the State Secretariat for Education, Research and
Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom
Space Agency. A complete and detailed list is available on the Euclid web site
(http://www.euclid-ec.org).
The first group of authors (up to and including M. Weiler) worked directly on
this paper. The other authors made substantial, multi-year contributions to
the Euclid project that enabled this paper in the first place.
The authors at MPIA acknowledge funding by the German Space Agency DLR under
grant numbers 50 OR 1202 and 50 QE 2003. The work by J.M. Carrasco and M.
Weiler was (partially) funded by the Spanish MICIN/AEI/10.13039/501100011033
and by “ERDF A way of making Europe” by the European Union through grant
RTI2018-095076-B-C21, and the Institute of Cosmos Sciences University of
Barcelona (ICCUB, Unidad de Excelencia ’María de Maeztu’) through grant
CEX2019-000918-M.
The authors thank Thomas Weber (OBJ; now Materion Balzers Optics) for the
technical support with the transmission measurements of the NISP filter
substrates, and the anonymous referee for their useful comments.
The plots in this publication were prepared with TOPCAT (Taylor 2005) and
Matplotlib (Hunter 2007).

## References

  * Amra et al. (2021) Amra, C., Lequime, M., & Zerrad, M. 2021, Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities (Cambridge University Press)
  * Astropy Collaboration: Price-Whelan et al. (2018) Astropy Collaboration: Price-Whelan, A. M., Sipőcz, B. M., Günther, H. M., et al. 2018, AJ, 156, 123
  * Astropy Collaboration: Robitaille et al. (2013) Astropy Collaboration: Robitaille, T. P., Tollerud, E. J., Greenfield, P., et al. 2013, A&A, 558, A33
  * Bai et al. (2018) Bai, Y., Farris, M., Fischer, L., et al. 2018, in High Energy, Optical, and Infrared Detectors for Astronomy VIII, ed. A. D. Holland & J. Beletic, Vol. 10709, International Society for Optics and Photonics (SPIE), 261 – 275
  * Bodendorf et al. (2019) Bodendorf, C., Geis, N., Grupp, F., et al. 2019, in Astronomical Optics: Design, Manufacture, and Test of Space and Ground Systems II, ed. T. B. Hull, D. W. Kim, & P. Hallibert, Vol. 11116, International Society for Optics and Photonics (SPIE), 257 – 278
  * Bohlin (2014) Bohlin, R. C. 2014, AJ, 147, 127
  * Bohlin & Deustua (2019) Bohlin, R. C. & Deustua, S. E. 2019, AJ, 157, 229
  * Bohlin et al. (2020) Bohlin, R. C., Hubeny, I., & Rauch, T. 2020, AJ, 160, 21
  * Bruzual & Charlot (2003) Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000
  * Calzetti et al. (2000) Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682
  * Carpenter (2001) Carpenter, J. M. 2001, AJ, 121, 2851
  * Chiggiato (2020) Chiggiato, P. 2020, arXiv e-prints, arXiv:2006.07124
  * Cropper et al. (2012) Cropper, M., Cole, R., James, A., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, ed. M. C. Clampin, G. G. Fazio, H. A. MacEwen, & J. Oschmann, Jacobus M., 84420V
  * Crouzet et al. (2020) Crouzet, P.-E., Shortt, B., Beaufort, T., et al. 2020, in X-Ray, Optical, and Infrared Detectors for Astronomy IX, ed. A. D. Holland & J. Beletic, Vol. 11454, International Society for Optics and Photonics (SPIE), 43 – 60
  * Cutri et al. (2003) Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online Data Catalog, II/246
  * Engelhart et al. (2017) Engelhart, D. P., Cooper, R., Cowardin, H., et al. 2017, in Advanced Maui Optical and Space Surveillance (AMOS) Technologies Conference, ed. S. Ryan, 21
  * Euclid Collaboration: Barnett et al. (2019) Euclid Collaboration: Barnett, R., Warren, S. J., Mortlock, D. J., et al. 2019, A&A, 631, A85
  * Euclid Collaboration: Blanchard et al. (2020) Euclid Collaboration: Blanchard, A., Camera, S., Carbone, C., et al. 2020, A&A, 642, A191
  * Euclid Collaboration: Ilbert et al. (2021) Euclid Collaboration: Ilbert, O., de la Torre, S., Martinet, N., et al. 2021, A&A, 647, A117
  * Euclid Collaboration: Scaramella et al. (2021) Euclid Collaboration: Scaramella, R., Amiaux, J., Mellier, Y., et al. 2021, arXiv e-prints, arXiv:2108.01201
  * Fang et al. (2019) Fang, Y., Furniss, D., Jayasuriya, D., et al. 2019, Optical Materials: X, 2, 100030
  * Gaia Collaboration: Eyer et al. (2019) Gaia Collaboration: Eyer, L., Rimoldini, L., Audard, M., et al. 2019, A&A, 623, A110
  * Gaia Collaboration: Prusti et al. (2016) Gaia Collaboration: Prusti, T., de Bruijne, J. H. J., Brown, A. G. A., et al. 2016, A&A, 595, A1
  * González-Fernández et al. (2017) González-Fernández, C., Hodgkin, S. T., Irwin, M. J., et al. 2017, MNRAS, 474, 5459
  * Green (2001) Green, D. B. 2001, Satellite Contamination and Materials Outgassing Knowledgebase - An Interactive Database Reference, NASA STI/Recon Technical Report N
  * Grün et al. (1985) Grün, E., Zook, H., Fechtig, H., & Giese, R. 1985, Icarus, 62, 244
  * Grupp et al. (2019) Grupp, F., Kaminski, J., Bodendorf, C., et al. 2019, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11116, Astronomical Optics: Design, Manufacture, and Test of Space and Ground Systems II, 1111618
  * Hanisch et al. (2001) Hanisch, R. J., Farris, A., Greisen, E. W., et al. 2001, A&A, 376, 359
  * Hewett et al. (2006) Hewett, P. C., Warren, S. J., Leggett, S. K., & Hodgkin, S. T. 2006, MNRAS, 367, 454
  * Hodgkin et al. (2009) Hodgkin, S. T., Irwin, M. J., Hewett, P. C., & Warren, S. J. 2009, MNRAS, 394, 675
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90
  * Ilbert et al. (2009) Ilbert, O., Capak, P., Salvato, M., et al. 2009, ApJ, 690, 1236
  * Korsch (1977) Korsch, D. 1977, Appl. Opt., 16, 2074
  * Laureijs et al. (2011) Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv e-prints, arXiv:1110.3193
  * Leggett et al. (2006) Leggett, S. K., Currie, M. J., Varricatt, W. P., et al. 2006, MNRAS, 373, 781
  * Löfdahl et al. (2011) Löfdahl, M. G., Henriques, V. M. J., & Kiselman, D. 2011, A&A, 533, A82
  * Maciaszek et al. (2016) Maciaszek, T., Ealet, A., Jahnke, K., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9904, Space Telescopes and Instrumentation 2016: Optical, Infrared, and Millimeter Wave, ed. H. A. MacEwen, G. G. Fazio, M. Lystrup, N. Batalha, N. Siegler, & E. C. Tong, 99040T
  * Oke & Gunn (1983) Oke, J. B. & Gunn, J. E. 1983, ApJ, 266, 713
  * Pelizzo et al. (2021) Pelizzo, M. G., Corso, A. J., Santi, G., et al. 2021, Scientific Reports, 11, 3429
  * Pickles & Depagne (2010) Pickles, A. & Depagne, É. 2010, PASP, 122, 1437
  * Polletta et al. (2007) Polletta, M., Tajer, M., Maraschi, L., et al. 2007, ApJ, 663, 81
  * Pons et al. (2019) Pons, E., McMahon, R. G., Simcoe, R. A., et al. 2019, MNRAS, 484, 5142
  * Prevot et al. (1984) Prevot, M. L., Lequeux, J., Maurice, E., Prevot, L., & Rocca-Volmerange, B. 1984, A&A, 132, 389
  * Prieto et al. (2012) Prieto, E., Amiaux, J., Auguères, J.-L., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, ed. M. C. Clampin, G. G. Fazio, H. A. MacEwen, & J. Oschmann, Jacobus M., 84420W
  * Rodmann et al. (2019) Rodmann, J., Miller, A., Bunte, K. D., & Millinger, M. 2019, in First International Orbital Debris Conference, Vol. 2109, 6070
  * Rogalski (2005) Rogalski, A. 2005, Reports on Progress in Physics, 68, 2267
  * Secroun et al. (2018) Secroun, A., Barbier, R., Buton, C., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10709, High Energy, Optical, and Infrared Detectors for Astronomy VIII, ed. A. D. Holland & J. Beletic, 1070921
  * Sheikh et al. (2008) Sheikh, D. A., Connell, S. J., & Dummer, R. S. 2008, in Space Telescopes and Instrumentation 2008: Optical, Infrared, and Millimeter, ed. J. M. O. Jr., M. W. M. de Graauw, & H. A. MacEwen, Vol. 7010, International Society for Optics and Photonics (SPIE), 1277 – 1281
  * Simonetto et al. (2020) Simonetto, F., Marmonti, M., & Potenza, M. A. C. 2020, Journal of Astronomical Telescopes, Instruments, and Systems, 6, 038004
  * Smith (2008) Smith, W. 2008, Modern Optical Engineering: The Design of Optical Systems, McGraw Hill professional (McGraw-Hill Professional)
  * Sun et al. (2020) Sun, X., Abshire, J., Lauenstein, J.-M., et al. 2020, IEEE Transactions on Nuclear Science, PP, 1
  * Sutherland et al. (2015) Sutherland, W., Emerson, J., Dalton, G., et al. 2015, A&A, 575, A25
  * Tallada et al. (2020) Tallada, P., Carretero, J., Casals, J., et al. 2020, Astronomy and Computing, 32, 100391
  * Tan & Arndt (2000) Tan, C. & Arndt, J. 2000, Journal of Physics and Chemistry of Solids, 61, 1315
  * Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
  * Tonry et al. (2018) Tonry, J. L., Denneau, L., Flewelling, H., et al. 2018, ApJ, 867, 105
  * Venancio et al. (2020) Venancio, L. M. G., Carminati, L., Amiaux, J., et al. 2020, in Space Telescopes and Instrumentation 2020: Optical, Infrared, and Millimeter Wave, ed. M. Lystrup, M. D. Perrin, N. Batalha, N. Siegler, & E. C. Tong, Vol. 11443, International Society for Optics and Photonics (SPIE), 45 – 60
  * Waczynski et al. (2016) Waczynski, A., Barbier, R., Cagiano, S., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9915, High Energy, Optical, and Infrared Detectors for Astronomy VII, ed. A. D. Holland & J. Beletic, 991511

##  Appendix A Passband variations

Table 5: Coefficients for Eq. (8) to compute the blueshifted cut-on and cut-off wavelengths $\lambda_{50}^{\rm on/off}(z,y)$ of the filter flanks in the focal plane’s R_mosaic coordinate system (Sect. 4.2). We use the e-notation to simplify a software implementation. The last column, ‘Test’, can be used to verify computer code for $z=30$, $y=50$. The corresponding blueshift of the passbands is shown in Fig. 8. Flank | $a_{0}$ | $b_{1}$ | $b_{2}$ | $b_{3}$ | $c_{1}$ | $c_{2}$ | $c_{3}$ | Test
---|---|---|---|---|---|---|---|---
$Y_{\scriptscriptstyle\rm E}$ cut-on | 949.58 |  4.21895e$-$04 |  $-$1.86149e$-$04 | 8.59708e-10 |  3.83384e$-$03 |  $-$1.83650e$-$04 |  $-$4.81958e$-$08 | 949.15
$Y_{\scriptscriptstyle\rm E}$ cut-off | 1212.22 |  $-$1.14852e$-$03 |  $-$2.32133e$-$04 |  $-$6.27018e$-$09 |  4.13559e$-$03 |  $-$2.28732e$-$04 |  $-$5.85140e$-$08 | 1211.60
$J_{\scriptscriptstyle\rm E}$ cut-on | 1167.61 |  $-$2.53327e$-$03 |  $-$2.36385e$-$04 |  $-$1.71028e$-$08 |  4.90696e$-$03 |  $-$2.37536e$-$04 |  $-$5.19171e$-$08 | 1166.97
$J_{\scriptscriptstyle\rm E}$ cut-off | 1566.94 |  $-$1.57482e$-$03 |  $-$3.12973e$-$04 |  $-$8.43209e$-$09 |  5.60578e$-$03 |  $-$3.07454e$-$04 |  $-$8.81555e$-$08 | 1566.11
$H_{\scriptscriptstyle\rm E}$ cut-on | 1521.51 |  6.95575e$-$05 |  $-$3.00405e$-$04 |  $-$4.34174e$-$09 |  8.25013e$-$03 |  $-$2.83213e$-$04 |  $-$7.85756e$-$08 | 1520.94
$H_{\scriptscriptstyle\rm E}$ cut-off | 2021.30 |  $-$8.28076e$-$05 |  $-$3.91185e$-$04 |  $-$7.98574e$-$09 |  1.14732e$-$02 |  $-$3.67085e$-$04 |  $-$1.02239e$-$07 | 2020.59
Figure 15: Maximum magnitude difference for galaxies in the Euclid SC8 catalog due to passband variations, when a source moves from the center of the FPA to the corner. Note the log-scaling of the y-axis, the effect is typically on the order of a few milli-mag (see Table 6). Figure 16: Same as Fig. 15, but for stars. Table 6: Mean and RMS (in mag) of the distributions shown in Figs. 15 and 16. | Galaxies | Stars
---|---|---
Filter | Mean | RMS | Mean | RMS
$Y_{\scriptscriptstyle\rm E}$ | 0.0047 | 0.0052 | 0.0004 | 0.0020
$J_{\scriptscriptstyle\rm E}$ | 0.0027 | 0.0028 | 0.0009 | 0.0013
$H_{\scriptscriptstyle\rm E}$ | 0.0019 | 0.0024 | -0.00142 | 0.00074

##  Appendix B Transformation from and to the NISP photometric system for
galaxies

Equations (21) through (33) convert NISP AB magnitudes to 2MASS, MKO and VISTA
AB magnitudes for galaxies. Before using these transformations, read Sect.
6.2.2. See also Fig. 17.

$\displaystyle J_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.005+0.314\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.037)$ (21)
$\displaystyle H_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.004+0.243\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.047)$ (22)
$\displaystyle K_{\rm s,\scriptscriptstyle\rm 2MASS}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.001-0.590\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.101)$ (23)
$\displaystyle Z_{\scriptscriptstyle\rm MKO}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.070+2.072\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.204)$ (24)
$\displaystyle Y_{\scriptscriptstyle\rm MKO}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle-0.002+1.234\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.068)$ (25)
$\displaystyle J_{\scriptscriptstyle\rm MKO}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.009+0.286\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.053)$ (26)
$\displaystyle H_{\scriptscriptstyle\rm MKO}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.005+0.264\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.044)$ (27)
$\displaystyle K_{\scriptscriptstyle\rm MKO}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle+0.004-0.635\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.100)$ (28)
$\displaystyle Z_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.055+2.224\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.225)$ (29)
$\displaystyle Y_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.000+1.282\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.073)$ (30)
$\displaystyle J_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.008+0.282\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.051)$ (31)
$\displaystyle H_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.004+0.244\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.044)$ (32)
$\displaystyle K_{\rm s,\scriptscriptstyle\rm VISTA}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.001-0.569\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.098)$ (33)

Equations (34) through (42) convert 2MASS, MKO and VISTA AB magnitudes to NISP
AB magnitudes for galaxies. Before using these transformations, read Sect.
6.2.2. See also Fig. 18.

$\displaystyle Y_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle+0.052+1.479\;(J_{\scriptscriptstyle\rm
2MASS}-H_{\scriptscriptstyle\rm 2MASS})$ $\displaystyle\quad(\sigma=0.105)$
(34) $\displaystyle Y_{\scriptscriptstyle\rm E}$
$\displaystyle=J_{\scriptscriptstyle\rm MKO}$
$\displaystyle+0.014+0.703\;(Y_{\scriptscriptstyle\rm
MKO}-J_{\scriptscriptstyle\rm MKO})$ $\displaystyle\quad(\sigma=0.047)$ (35)
$\displaystyle Y_{\scriptscriptstyle\rm E}$
$\displaystyle=J_{\scriptscriptstyle\rm VISTA}$
$\displaystyle+0.015+0.659\;(Y_{\scriptscriptstyle\rm
VISTA}-J_{\scriptscriptstyle\rm VISTA})$ $\displaystyle\quad(\sigma=0.049)$
(36) $\displaystyle J_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle-0.005+0.676\;(J_{\scriptscriptstyle\rm
2MASS}-H_{\scriptscriptstyle\rm 2MASS})$ $\displaystyle\quad(\sigma=0.037)$
(37) $\displaystyle J_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm MKO}$
$\displaystyle+0.005+0.652\;(J_{\scriptscriptstyle\rm
MKO}-H_{\scriptscriptstyle\rm MKO})$ $\displaystyle\quad(\sigma=0.042)$ (38)
$\displaystyle J_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm VISTA}$
$\displaystyle+0.004+0.665\;(J_{\scriptscriptstyle\rm
VISTA}-H_{\scriptscriptstyle\rm VISTA})$ $\displaystyle\quad(\sigma=0.042)$
(39) $\displaystyle H_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle+0.000-0.192\;(J_{\scriptscriptstyle\rm
2MASS}-H_{\scriptscriptstyle\rm 2MASS})$ $\displaystyle\quad(\sigma=0.057)$
(40) $\displaystyle H_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm MKO}$
$\displaystyle-0.002-0.207\;(J_{\scriptscriptstyle\rm
MKO}-H_{\scriptscriptstyle\rm MKO})$ $\displaystyle\quad(\sigma=0.055)$ (41)
$\displaystyle H_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm VISTA}$
$\displaystyle-0.002-0.192\;(J_{\scriptscriptstyle\rm
VISTA}-H_{\scriptscriptstyle\rm VISTA})$ $\displaystyle\quad(\sigma=0.054)$
(42)

###  B.1 About the substructure in Figs. 17 and 18

Some of the color-color plots in Figs. 17 and 18 show substructure that appear
artificial. Remember that these are not real observations, but the SC8
synthetic SED templates, redshifted between $z=0.0$ and $2.3$. Some of the
redshifted templates distribute fairly widely over the color-color space,
whereas others – mostly star-forming galaxies – remain more concentrated. The
latter then visually dominate the appearance of the linearly scaled color-
color density maps. We choose a linear scaling so that the causal connection
between the data and the linear fits (red lines) becomes evident. A
logarithmic scaling is misleading in most cases, insinuating a poor fit
quality.

As an example for substructure, consider the bottom two rows of Fig. 18,
showing the difference in the NISP $H_{\scriptscriptstyle\rm E}$-band and
VISTA $H$-band magnitudes on the y-axis. Two parallel main branches are seen,
mostly from star-forming galaxies. This is a consequence of the NISP
$H_{\scriptscriptstyle\rm E}$-band being nearly twice as wide as the VISTA
$H$-band (see Fig. 1). The two bands respond differently to spectra with
specific continuum shapes and emission lines, also depending on redshift. The
small bridge connecting the two branches represents starbursts with a specific
metallicity and extinction evolving from redshift $z=0.25$ to $0.55$. The true
complexity of this specific color-color space is shown in Fig. 19. This also
illustrates that the linear transformation equations may fail catastrophically
for individual galaxies.

Figure 17: Magnitude differences for galaxies between external systems (2MASS,
MKO, VISTA) and NISP, as a function of NISP color. The red lines show the
linear fits given in Eqs. (21) through (33). Figure 18: Magnitude differences
for galaxies between NISP and external systems (2MASS, MKO, VISTA), as a
function of color in an external system. The red lines show the linear fits
given in Eqs. (34) through (42). Figure 19: Zoom into the lower left panel of
Fig. 18, showing more data and details about how the redshifted Euclid SC8
galaxy SEDs populate this particular color-color space. The contours trace the
point density with a logarithmic spacing. Template numbers $0$–$7$ correspond
to elliptical and lenticular types, templates $8$–$11$ to Hubble types Sa and
Sb, templates $12$–$18$ to types Sc and Sd, and templates $18$-$30$ to
starbursts of various age and metallicity.

##  Appendix C Transformation from and to the NISP photometric system for
stars (spectral types O to K)

Equations (43) through (55) convert NISP AB magnitudes to 2MASS, MKO and VISTA
AB magnitudes for stars of spectral types O to K. The $1\,\sigma$ residuals
are given in parentheses. Before using these transformations, read Sect.
6.2.2. See also Fig. 20.

$\displaystyle J_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.010+0.423\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.016)$ (43)
$\displaystyle H_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.058+0.147\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.021)$ (44)
$\displaystyle K_{\rm s,\scriptscriptstyle\rm 2MASS}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle+0.239-0.356\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.041)$ (45)
$\displaystyle Z_{\scriptscriptstyle\rm MKO}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.082+2.018\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.107)$ (46)
$\displaystyle Y_{\scriptscriptstyle\rm MKO}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle-0.012+1.115\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.020)$ (47)
$\displaystyle J_{\scriptscriptstyle\rm MKO}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.012+0.406\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.021)$ (48)
$\displaystyle H_{\scriptscriptstyle\rm MKO}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.055+0.184\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.018)$ (49)
$\displaystyle K_{\scriptscriptstyle\rm MKO}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle+0.271-0.333\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.045)$ (50)
$\displaystyle Z_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle+0.151+1.891\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.113)$ (51)
$\displaystyle Y_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle-0.011+1.165\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.025)$ (52)
$\displaystyle J_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.012+0.406\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.020)$ (53)
$\displaystyle H_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.055+0.162\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.018)$ (54)
$\displaystyle K_{\rm s,\scriptscriptstyle\rm VISTA}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle+0.228-0.340\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.036)$ (55)

Equations (56) through (64) convert from 2MASS, MKO and VISTA AB magnitudes to
NISP AB magnitudes for stars of spectral types O to K. Before using these
transformations, read Sect. 6.2.2. See also Fig. 21.

$\displaystyle Y_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle-0.005+1.134\;(J_{\scriptscriptstyle\rm
2MASS}-H_{\scriptscriptstyle\rm 2MASS})$ $\displaystyle\quad(\sigma=0.029)$
(56) $\displaystyle Y_{\scriptscriptstyle\rm E}$
$\displaystyle=J_{\scriptscriptstyle\rm MKO}$
$\displaystyle+0.008+0.789\;(Y_{\scriptscriptstyle\rm
MKO}-J_{\scriptscriptstyle\rm MKO})$ $\displaystyle\quad(\sigma=0.013)$ (57)
$\displaystyle Y_{\scriptscriptstyle\rm E}$
$\displaystyle=J_{\scriptscriptstyle\rm VISTA}$
$\displaystyle+0.005+0.748\;(Y_{\scriptscriptstyle\rm
VISTA}-J_{\scriptscriptstyle\rm VISTA})$ $\displaystyle\quad(\sigma=0.015)$
(58) $\displaystyle J_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle-0.007+0.786\;(J_{\scriptscriptstyle\rm
2MASS}-H_{\scriptscriptstyle\rm 2MASS})$ $\displaystyle\quad(\sigma=0.020)$
(59) $\displaystyle J_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm MKO}$
$\displaystyle-0.009+0.771\;(J_{\scriptscriptstyle\rm
MKO}-H_{\scriptscriptstyle\rm MKO})$ $\displaystyle\quad(\sigma=0.023)$ (60)
$\displaystyle J_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm VISTA}$
$\displaystyle-0.009+0.775\;(J_{\scriptscriptstyle\rm
VISTA}-H_{\scriptscriptstyle\rm VISTA})$ $\displaystyle\quad(\sigma=0.022)$
(61) $\displaystyle H_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle+0.069-0.120\;(J_{\scriptscriptstyle\rm
2MASS}-H_{\scriptscriptstyle\rm 2MASS})$ $\displaystyle\quad(\sigma=0.021)$
(62) $\displaystyle H_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm MKO}$
$\displaystyle+0.069-0.145\;(J_{\scriptscriptstyle\rm
MKO}-H_{\scriptscriptstyle\rm MKO})$ $\displaystyle\quad(\sigma=0.017)$ (63)
$\displaystyle H_{\scriptscriptstyle\rm E}$
$\displaystyle=H_{\scriptscriptstyle\rm VISTA}$
$\displaystyle+0.066-0.131\;(J_{\scriptscriptstyle\rm
VISTA}-H_{\scriptscriptstyle\rm VISTA})$ $\displaystyle\quad(\sigma=0.018)$
(64)

Figure 20: Magnitude differences for stars of spectral types O to K between
external systems (2MASS, MKO, VISTA) and NISP, as a function of NISP color.
Data are taken from the Euclid SC8 simulation pilot catalogs. The red lines
show the linear fits given in Eqs. (43) through (55). Figure 21: Magnitude
differences for stars of spectral types O to K between NISP and external
systems (2MASS, MKO, VISTA), as a function of color in an external system. The
red lines show the linear fits given in Eqs. (56) through (64).

##  Appendix D Transformation from and to the NISP photometric system for
stars (MLT types)

Equations (65) through (77) convert NISP AB magnitudes to 2MASS, MKO and VISTA
AB magnitudes for MLT types. The $1\,\sigma$ residuals are given in
parentheses. Before using these transformations, read Sect. 6.2.2. See also
Fig. 22.

$\displaystyle J_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.051+1.248\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.034)$ (65)
$\displaystyle H_{\scriptscriptstyle\rm 2MASS}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.183+0.226\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.051)$ (66)
$\displaystyle K_{\rm s,\scriptscriptstyle\rm 2MASS}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.006-0.612\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.082)$ (67)
$\displaystyle Z_{\scriptscriptstyle\rm MKO}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+1.032+1.877\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.289)$ (68)
$\displaystyle Y_{\scriptscriptstyle\rm MKO}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle+0.511+1.821\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.096)$ (69)
$\displaystyle J_{\scriptscriptstyle\rm MKO}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.239+1.469\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.092)$ (70)
$\displaystyle H_{\scriptscriptstyle\rm MKO}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.148+0.223\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.044)$ (71)
$\displaystyle K_{\scriptscriptstyle\rm MKO}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle+0.027-0.646\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.085)$ (72)
$\displaystyle Z_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=J_{\scriptscriptstyle\rm E}$
$\displaystyle+0.979+2.368\;(Y_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.376)$ (73)
$\displaystyle Y_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle+0.596+1.758\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.096)$ (74)
$\displaystyle J_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.212+1.429\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.081)$ (75)
$\displaystyle H_{\scriptscriptstyle\rm VISTA}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle-0.152+0.204\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.043)$ (76)
$\displaystyle K_{\rm s,\scriptscriptstyle\rm VISTA}$
$\displaystyle=H_{\scriptscriptstyle\rm E}$
$\displaystyle+0.017-0.594\;(J_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E})$ $\displaystyle\quad(\sigma=0.083)$ (77)