TP aberrations lab sessions 2016

LEns

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Optics Labwork - Semester 2

AberrationsGeneral presentation

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1The point source method . . . . . . . . . . . . . . . . . . . . . . 4Wavefront measurements . . . . . . . . . . . . . . . . . . . . . 6Specifications of the lenses . . . . . . . . . . . . . . . . . . . . 12Results sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

The point source methodLab 1 On-axis aberrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Lab 2 Off-axis aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Wavefront mesurementsLab 3 Zygo wavefront analysis by interferometric

measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Lab 4 Haso Shack-Hartmann wavefront analysis . . . . 53

Engineer 2nd year - PalaiseauVersion : 6 janvier 2016Year 2015-2016

Catherine BURCKLENDavid HOLLEVILLELionel JACUBOWIEZ

Gaëlle LUCAS-LECLIN

Rules

All the information about Travaux Pratiques (TP) is posted on thebulletin board on the first floor of the building, next to the entrance to

the TP.Look at the board regularly.

Attendance

Attendance is mandatory for all lab sessions. If you or your labpartner has a major problem preventing their attendance, the othershould nevertheless attend the session and do the lab. In addi-tion, for Optics labs, each partner will then write an individual report.

Justification for non attendance The student who cannot attendwill both have to deposit a written justification at the secretary ?soffice, and also warn directly the teacher in charge of the stu-dents ? lab of the reason for his/her non attendance (before thesession, if the absence is predictable).

Justified absence. Coming back to do the missed lab The studentmust always contact the lab session instructors to envisage thepossibility of coming back to do the missed lab. This possibilitywill depend on the availability of the instructor, the equipmentand the lab rooms. If agreed, the student then comes back to dothe lab and :In optics labs, the student writes a specific lab report that will

get graded. If it is not possible to find a date for this rerunlab on behalf of the availability of the students ? lab service,then this lab will not be redone nor graded (the average forthe labs will be done using all the other sessions). This labwill still remain part of the programme for the final lab exam

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and the student might still end up having to perform this labon the day of the exam.

In ETI and ProTIS, the synthesis of the relevant thËme, writ-ten by the two lab partners, will have to include the resultsof the two individual sessions (the regular one and the rerunone)

If the student refuses the proposed date for the rerun lab, he willbe considered as absent without justification.

Absence without a proper justification All absence without jus-tification will lead to :in Optics, a grade of zero for the missed session and the im-

possibility to work on this particular lab before the revisionsessions. In case of repeted absences, the teacher in charge ofthat school year (1st, 2nd or 3rd ) will not allow the student toattend the first lab exam session organized at the end of theschool year.

In ETI and ProTIS, a grade of zero will be given for the corres-ponding synthesis grade.

Punctuality

You are expected to arrive on time for the lab sessions. A student whois very or frequently late will not be allowed to complete the lab andthe session will be considered as an unexcused absence (see above).

Plagiarism

Plagiarism is the inclusion in your own work, that of someone else(text, drawings, photos, data etc.) without clear attribution. Examplesinclude :

— Copying word for word (or nearly so) from a book or Web pagewithout putting the passage in quotes and mentioning its source.

— Inserting in your work graphs or images coming from externalsources (excepting the TP documents) without indicating the source.

— Using the work of another student and presenting it as one’s own(even if the other student agrees).

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— Summarizing an original idea of another author using his or herwords without attribution.

— Partially or wholly translating a text without giving the source.Any student having demonstrably plagiarized a lab report will re-

ceive a grade 0/20 on that report and will be disciplined according tothe "règlement intérieur" (school guidlines).

Respect for the lab space and equipment

The student lab contains a large variety of high quality equipment.Please take good care of it.The equipment is often fragile and sensitive to dust. It is thereforestrictly forbidden to bring food or drink into the entire lab area, in-cluding the hallways.Take care to keep the area clean (if for example, your shoes are dirty,leave them at the entrance).

If for any reason you need access to the labs outside of the regularlab hours, see the TP staff , Thierry Avignon or Cédric Lejeune (officeS1.18).

Évaluation de votretravail

TP & compte-rendusBarème indicatif sur 20 points :

Si la présentation oralen’a pu avoir lieu :

Habilité en manipulation 5 5Compte-rendu 10 15Présentation orale 5

Comptes-rendus :Sauf indication contraire, les comptes-rendus doivent être déposés

sur le site Libres Savoirs une semaine après la séance. Merci de respec-ter les consignes suivantes :

— Certifiez l’originalité de votre travail en faisant figurer la men-tion : Nous attestons que ce travail est original, que nous citonsen référence toutes les sources utilisées et qu’il ne comporte pasde plagiat.

— Vérifiez que vos noms et le numéro de votre binôme fi-gurent sur la première page de votre compte-rendu avantde la transformer en .pdf.

— Renommez le fichier .pdf selon le format :G4B05DupondEtDupontTPmachintrucsuper.pdf avant de ledéposer sur le site.

Attention : un point de moins par jour de retard !

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Chaque vendredi, la liste des éventuels compte-rendus manquantsest affichée sur le panneau du LEnsE, pensez à la vérifier !

Présentation orale :

La présentation orale a plusieurs objectifs :— elle nous permet de vérifier si vous avez compris les points essen-

tiels du TP, si vous êtes capable de prendre du recul par rapportaux manipulations effectuées.

— elle nous permet de corriger les points que vous auriez mal com-pris.

— elle vous entraîne à présenter oralement un travail expérimentalde manière synthétique. C’est une situation que vous rencontre-rez souvent dans votre vie professionnelle.

La présentation orale aura lieu durant le TP, sa durée est de 5 mi-nutes pour un binôme, 7 minutes pour un trinôme, pendant lesquelleschaque membre du binôme ou trinôme doit intervenir.

Elle s’adresse à un "opticien de passage" (par exemple un ancienélève de SupOptique sorti il y a plus de 5 ans) qui a priori ne connaîtrien à la manipulation. Vous pouvez choisir le plan qui vous semblele mieux adapté, utiliser le matériel du TP et montrer les résultatsobtenus.

Seront évaluées :— les qualités pédagogiques (clarté, précision, enthousiasme,. . . )— les capacités de synthèse,— la qualité scientifique de la présentation,— la gestion du temps.Défauts à éviter absolument :— faire un résumé « historique et linéaire » du déroulement de la

séance,— nous présenter des points de détails (que nous vous avons juste-

ment expliqués pendant la séance parce qu’ils étaient délicats),— utiliser des termes trop spécialisés sans les expliquer.

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ExamenL’examen de fin d’année en 1A et 2A se déroule de la façon suivante

(tous les documents sont autorisés) :— L’élève tire un sujet au hasard. Celui-ci tient en quelques phrases :

caractériser les aberrations d’un objectif (2A), mesurer le bruitd’un système de détection (2A),...Le sujet est très proche d’un TPde l’année (premier ou deuxième semestre).

— L’élève a ensuite 2 heures pour :— réfléchir à la meilleure façon de réaliser la mesure qui lui est

demandée,— effectuer le montage et le réglage de la manipulation,— effectuer les mesures,— évaluer les incertitudes sur ses mesures,— analyser les résultats,— et préparer la présentation orale.

— Au bout des deux heures, lors d’un exposé oral d’environ 10 mi-nutes, l’élève explique :— comment il a effectué la mesure qui lui était demandée,— quelles sont les précautions particulières à prendre,— dans quel ordre faire les réglages,— et il commente les mesures effectuées.

— Les enseignants du jury posent ensuite des questions permettantde préciser ou d’approfondir le travail réalisé par l’élève.

En préparation de cet examen final, en plus du soin apporté auxcomptes-rendus pendant l’année et à votre participation active pendantles séances, nous ne pouvons que vous conseiller fortement de réviserbien à l’avance les travaux pratiques des deux semestres.

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GENERAL PRESENTATION OF THE

ABERRATIONS LAB SESSIONS

Updated January 2016

INTRODUCTION The goal of the following lab sessions is to study and analyze the defects of an optical system, i.e. the geometrical and chromatic aberrations. The lab sessions dedicated to this study should be considered as a whole, since you will study the same optical systems by various methods and then summarize and compare the results.

Preparation

Each lab session should be prepared ahead of time: for each lab session, we expect from you that you understand the measurement methods ahead of time, and that you do the preliminary calculations for the set-up specific to this lab session. Preparatory should be answered during this preparation phase.

Lab report

Each group will study a set of three optical systems, contained in a labeled box (A, B, C or D), and described below. Please note carefully the label of your set of optics, since each group has a different set of optics, and make sure that you always study the same set of optics from one lab session to the next one (do not mix optics and labels!). This will allow you to evaluate the performance of your set of optics by different methods and compare them.

The lab report associated with LAB SESSION N° 4 (HASO) should be handed in at the end of the corresponding session.

For LAB SESSIONS N°1-2-3, we ask you to give in a global lab report that synthesizes your results. In this report, you should analyze the performance of the optical systems according to

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relevant criteria (size of the image spot, optical path delay, modulation transfer function …) and compare the results obtained by the different methods available. The final report should be less than 24 pages, results forms included.

Use of on-line schematics or documents is permitted but should be referenced accurately. If this is not the case, you will endure a severe penalty.

Evaluation

At the end of each lab session you will perform a second oral presentation (10 minutes) where I will evaluate your capabilities to synthesize what you think is most relevant in your observations and results. This presentation will account for 25% of the final grade of your work on LAB SESSIONS N°1-2-3.

The evaluation on SESSIONS N°1-2-3 will also take into account:

your final lab report (50% of the final grade, based on a syntheses of the key points of the experiments, the presentation of the results, your analysis and synthesis of the results, and a comparison of the methods),

your work throughout the lab sessions (25% of the final grade, based on your autonomy, your understanding of the experiments, you ability to perform the required experiments within the duration of the lab-session (4:30 hours minus 15 minutes of oral presentations), and the quality of your measurements).

To optimize your performance, the lab sessions should thus be prepared ahead of time.

To be on time is not an option: the sessions start at 1:30pm and end at 6:00pm.

In practice

Each group will study the optics contained in a box (labeled A, B C or D) by different methods explored throughout LAB SESSIONS N°1-2-3: a plano-convex singlet, a Clairaut-Mossotti doublet, and a magnifying objective. You will study these lenses by two complementary approaches:

The point source method

Lab session n°1: spherical aberration and chromatism

Lab session n°2: off-axis aberrations

During these lab sessions, you will observe the image spot directly with a microscope viewer. Observing the image spot, i.e. the Point Spread Function of the optical system under test, allows you to evaluate the performance of the optical system rapidly on axis and off axis, in the presence of monochromatic or polychromatic light, even with large aberrations.

Analysis of the transmitted wavefront

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Lab session n°3: Fizeau interferometer (ZYGO)

Lab session n°4: Shack-Hartmann analyzer (HASO)

During these lab sessions, you will study the wavefront after transmission through the optical system under test, firstly by an interferometric method (with the ZYGO), and secondly by a geometrical method (with the HASO). Both methods allow you to measure the Optical Path Delay between the transmitted wavefront and the ideal spherical wavefront, i.e. the “reference sphere”. These two methods are well suited for optical systems with small aberrations.

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THE POINT SOURCE METHOD

An easy and fairly efficient way to study the quality of an optical system is to observe and measure the image of a point source, also called the Point Spread Function (PSF). The shape and the size of this image depend on the aberrations of the optical system. A perfect system is diffraction limited, i.e. the PSF is an Airy function. The measurements can be made with a microscope viewer and can be performed either with the eye or with a camera. Defocusing the observation plane slightly around the best focus plane gives much information about the aberrations of the system.

Figure 1: Experimental set-up

This method allows the study of an optical system on axis and also off axis, by moving the source in the object plane or by rotating the optical system under test around the image nodal point. If one wants to study the optical system in an infinite-focus conjugation, the point source has to be at a large distance with respect to the focal length of the optical system (LAB WORK N°1) or at the object focal point of a good quality collimator, with small numerical aperture (LAB WORK N°2).

Warning: It is crucial to systematically make sure that the pupil of the system under test is fully illuminated otherwise the aberrations would be grossly under estimated. If required, a condenser can be inserted between the light source and the point source to ensure that the optical system under test is fully illuminated.

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1. Diameter of the “point” source: how to choose the pinhole

Ideally, the point source should be infinitely small in order to measure the PSF size of the optical system under test. In practice, however, the finite size of the pinhole makes the observed image larger than the PSF. The observed image is actually the convolution of the PSF that we want to study by the geometrical image of the pinhole. The geometrical image of the pinhole should thus be kept small with respect to the PSF diameter. Typically, for an optical system with a circular pupil, the size of the PSF is larger than or equal to the diameter of the Airy spot. On the other hand, the observation may be difficult by lack of light if the pinhole is smaller than necessary. The diameter of the pinhole should thus be adapted to the aberrations of the system under test:

For small aberrations (lenses that are nearly diffraction limited), the geometrical image of the pinhole should have a maximum diameter about 5 times smaller that the radius of the Airy spot in the image plane, if the pinholes available and the sensitivity of the detector allow it. A factor 10 is even better, if possible.

For large aberrations, choose a pinhole with a larger diameter to compensate for the spreading of light, whilst keeping the pinhole geometrical image smaller than the dimension of the PSF.

2. Experimental set up

To study an optical system that conjugates two planes at finite distances, you should use a pinhole at the correct distance from the lens under test.

For an infinite – focus conjugation, there are 2 possibilities:

Put the pinhole at a large distance with respect to the focal length of the optical system (LAB SESSION N°1). A distance of 10 times the focal length is typically sufficient.

Or, put the pinhole at the object focal point of a good quality collimator, with a pupil larger than the pupil of the lens under test (LAB SESSION N°2).

3. Choice of the objective of the microscope viewer

Numerical aperture of the microscope objective: the system under test must define the pupil of the whole set-up. Therefore, the numerical aperture of the microscope objective must be larger than the numerical aperture of the system under test.

Microscope objective magnification ratio: the PSF must be sufficiently magnified to be observed easily. This will automatically be the case if the numerical aperture of the microscope objective is chosen according to the previous paragraph. However, the size of the PSF should not exceed the full field of view of the microscope viewer. You should thus start with a moderate magnification ratio, and increase it progressively if necessary.

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WAVEFRONT MEASUREMENTS Both the ZYGO and the HASO sensors lead to a measurement of the wavefront defect, but they use two different methods. The ZYGO, which is based on the Fizeau interferometer, uses an interferometric approach, while the HASO approach, based on the Shack-Hartmann analysis, is a geometrical measurement. In both cases, the data processing allows a quantitative analysis of PSF (contributions of the various aberrations) and MTF of the tested optical system.

o ZYGO : interferometric measurement of a wavefront:

The Zygo analyzes interferences between a reference plane wavefront that has been reflected by a glass reference plane, and a plane wavefront that has propagated forth and back through the objective. The back-reflection is provided by a spherical reference mirror. The center of curvature of the spherical mirror is on the objective best focus.

o HASO : geometrical measurement of a wavefront:

Following the Shack-Hartmann method, an array of micro-lenses samples the wavefront and enables the measurement of the local slopes of the wavefront. The software reconstructs the wavefront in real time from the slopes measurements. This is very useful for alignment of the optical system under test. By comparison with a ZYGO interferometer, a Shack-Hartmann analyzer has a lower spatial resolution. It is not suited for optical systems with a large image numerical aperture.

Both the ZYGO and the HASO measure the wavefront, from which the software calculates various features such as the Point Spread Function, which you can compare directly with your visual measurements (see lab sessions based on the point source method).

1. Decomposition of the wavefront on the Zernike polynomials basis

It is possible to evaluate the various geometrical aberrations contribution by decomposing the wavefront defect on the Zernike polynomials orthogonal basis, i.e. by a least squares interpolation method (see Table 1 below for a definition of the Zernike polynomials ),u(Pi used by the ZYGO

software). Note that the polynomials ),u(Pi do not have the same norm. they can be normalized by

multiplying them by the factors ik indicated in Table 1, i.e. the polynomials ),u(Pk),u(Z iii constitute

an orthonormal basis over the disc pupil, with 2

i ),u(Z .

The software calculates the coefficients Ci of the following decomposition of the Optical Path Difference (OPD) over the circular pupil:

),u(PC ii

i

The RMS OPD is thus evaluated as follows:

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0i

2i

2i

kC

The Zernike polynomials basis is well adapted to describe aberrations over circular pupils only. Note that the HASO allows you to decompose the wavefront over the Legendre polynomials basis, which is better adapted for rectangular pupils.

The decomposition over the Seidel basis remains widely used for the 3rd order aberrations. The relationships between the Seidel amplitudes and the Zernike coefficients (see Table below) are the following:

Seidel Peak-to-Valley amplitude

Spherical aberration 8max C6 1 maxPV

Coma 27

26max CC3 maxPV 2

Astigmatism 25

24max CC maxPV 2

Polynomials used in ZyMOD /PC and HASO

Table 1: the first 36 Zernike polynomials ),u(Pi used in ZyMOD /PC and their associated coefficients ki in the decomposition of the wavefront on the Zernike basis. ‘u’ denotes the aperture variable, normalized to the edge of the pupil, and denotes the azimuth angle in the pupil.

Ci ki mcos)u(R),u(P ni

C0 1 1 Piston

C1 2 cosu

Tilt

C2 2 sinu

C3 3 1u2 2 Defocus

C4 6 2cosu2 Astigmatism

3rd order

C5 6 2sinu2

C6 8 cosu)2u3( 2 Coma

C7 8 sinu)2u3( 2

1 Note that if C15 is not negligible in the decomposition of the wave front, it is necessary to take it into account to evaluate correctly the spherical aberration of the 3rd order: max = 6 C8 – 30 C15. This can be extrapolated easily to the other terms.

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C8 5 1u6u6 24 Spherical aberration

C9 8 3cosu3 Trefoil

5th order

C10 8 3sinu3

C11 10 2cosu)3u4( 22 Astigmatism

C12 10 2sinu)3u4( 22

C13 12 cosu3u12u10 24 Coma

C14 12 sinu)3u12u10( 24

C15 7 1u12u30u20 246 Spherical aberration

C16 10 4cosu4 Tetrafoil

7th order

C17 10 4sinu4

C18 12 3cosu)4u5( 32 Trefoil

C19 12 3sinu)4u5( 32

C20 14 2cosu)6u20u15( 224 Astigmatism

C21 14 2sinu)6u20u15( 224

C22 16 cosu)4u30u60u35( 246 Coma

C23 16 sinu)4u30u60u35( 246

C24 9 1u20u90u140u70 2468 Spherical aberration

C25 12 5cosu5 Pentafoil

9th order

C26 12 5sinu5

C27 14 4cosu)5u6( 42 Tetrafoil

C28 14 4sinu)5u6( 42

C29 16 3cosu)10u30u21( 324 Trefoil

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C30 16 3sinu)10u30u21( 324

C31 18 2cosu)10u60u105u56( 2246 Astigmatism

C32 18 2sinu)10u60u105u56( 2246

C33 20 cosu)5u60u210u280u126( 2468 Coma

C34 20 sinu)5u60u210u280u126( 2468

C35 11 1u30u210u560u630u252 246810 Spherical aberration

C36 13 1u42u420u1680u3150u2772u924 24681012 Spherical aberration

11th

order

2. Simulation of the image spot

From the measured defects of the wavefront in the pupil, i.e. the Optical Path Differnce, both HASO and ZYGO could simulate the image spot of the optical system under test.

o Geometrical analysis: Spot diagram

Rays being perpendicular to the wavefront, it is straightforward to deduce the intersections of the rays with the image plane. This geometrical approach gives the Spot diagram. For a perfect system the spot diagram reduces to a single point (stigmatism).

o Fourier analysis: PSF and MTF

A second approach makes use of the diffraction theory to calculate the Point Spread Function (PSF) of the optical system, again starting from the measured wavefront. A wavefront analyzer measures the defects of the wavefront at every point of the pupil, pp y,x . Assuming that the pupil is uniformly illuminated, the amplitude of the

electromagnetic field in the pupil is related to the wavefront defects by:

)y,x(2j

0)y,x(j

0PP

pppp eaea)y,x(P

The incoherent Point Spread Function (the image spot), is calculated by Fourier Transform:

xr

xr

2PPPPPP y,xPFTy,xPy,xPFTrPSF

The PSF of a diffraction limited optical system with a circular pupil is an Airy spot (see Figure 2): it is the smallest image of a point source that an optical system can form. The diameter of the first dark ring of an Airy spot is:

10

1,22 sin .

The energy encircled within this first dark ring is 84% of the total power contained in the PSF.

Comparing the PSF of the system under test with the Airy spot limit is a good way to evaluate how important the defects of the system are. The Strehl ratio, RS , is the ratio between the maximum of the measured PSF and the maximum of the ideal (i.e. diffraction limited) PSF. The Maréchal criterion stipulates that, for visual observations, an optical system can be considered as diffraction limited if 8.0RS . In practice, values of RS below 0.2 are not relevant to describe the quality of an optical system.

Figure 2 : PSF of a diffraction limited lens with a numerical aperture of 0.08 at = 633 nm (left) and the corresponding encircled energy (right). Here, the PSF is an Airy spot.

The Modulation Transfer Function (MTF) is another way to evaluate the defects of an optical system that is illuminated by incoherent light. It is the Fourier Transform of the PSF.

rPSFFT

PPPPMTF

0

r

The Modulation Transfer Function yields the contrast of the image formed by the optical system when the object is a sinusoidal test pattern. The MTF is usually plotted versus the spatial frequency , measured in the image plane (see Figure 3).

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Figure 3 : MTF of a diffraction limited lens with a numerical aperture of 0.08 at = 633nm.

))(1)cos(Arc(2)(FTI 2

cccdiff

Comparing the MTF of the system under test with the MTF of a diffraction limited system with the same numerical aperture is also a good way to evaluate how much the aberrations of the system degrade the quality of the image. The cut-off frequency of a diffraction limited system is given by:

/sin2 maxc

where maxsin is the numerical aperture (NA) of the system under test, in the image space.

For c , 0MTF . In practice, contrasts below 20% yield an image of poor quality. For

systems with a circular pupil, the cut-off frequency at 20% is equal to c7.0 .

Note that the MTF contains exactly the same information as the PSF: one is merely the Fourier Transform of the other. It is just another way to evaluate the aberrations of an imaging system.

3. Important remark on the wavefront measurement

The wavefront defect measurement should be performed in a conjugate plane of the output pupil of the system. Otherwise, the propagation would lead to a deformation of the wavefront, which is particularly visible on the edge of the pupil when the focus adjustment is not accurate. The ZYGO has a CAM adjustment option (a motorized lens) that allows the output pupil of the optical system under test to be conjugated with the CCD plane where the interference pattern is recorded. In the case of the HASO labwork, this condition is not fulfilled. This may lead to measurement errors.

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SPECIFICATIONS OF THE LENSES

The specifications of the lenses that you will study are approximately the following:

LAB SESSIONS N°1-2-3

Plano-convex singlet

f’ = 150 mm (+/- 1%); = 25 mm; numerical aperture 08.0sin max

The characteristics of this singlet have been simulated with OSLO and are shown in Appendix 1.

Clairaut-Mossotti doublet (Thorlabs AC254-150-A1) f’ = 150 mm (+/- 1%); = 25 mm; 08.0sin max

There is no air space between the two lenses. The pair of glasses is BK7/SF5. The characteristics of this doublet have been simulated with OSLO and are shown in Appendix 2. It was designed to be aplanatic (i.e. no spherical aberration and no coma) and achromatic in the visible range.

Rodenstock magnifying objective

f’ = 150 mm (+/- 1%) – Iris diaphragm.

The characteristics of this objective will be measured precisely during the lab sessions. It was designed to magnify pictures taken with a 24mm x 36mm ISO200 film. The diameter of the grain of an ISO200 film is approximately 20µm.

LAB SESSION N°4

Doublet (Thorlab AC254-150-A1)

f’ = 150 mm, N = 6

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APPENDIX 1 : SIMULATED CHARACTERISTICS OF THE PLANO-CONVEX SINGLET, IN THE BEST ORIENTATION Glass: BK7, Effective focal length: 150 mm, Numerical aperture: 0.08 (Øe = 25 mm) Spot diagram on axis at = 633 nm The black circle represents the first dark ring of the Airy spot. Dimensions are in millimeters.

Paraxial focus plane

Minimum scatter focus plane

PSF on axis at = 633 nm

Paraxial focus plane

Minimum scatter focus plane

Longitudinal chromatism

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APPENDIX 2: SIMULATED CHARACTERISTICS OF THE

THORLABS DOUBLET AC254-150-A1

Effective focal length: 150 mm, Numerical aperture: 0.08 (Øe = 25 mm)

Surface data Doublet AC254-150-A1 SRF RADIUS THICKNESS APERTURE RADIUS GLASS SPE NOTE OBJ -- 1.0000e+20 8.7489e+18 AIR AST 91.620000 5.700000 12.500000 A N-BK7 C 2 -66.680000 2.200000 12.500000 P SF5 C 3 -197.700000 146.135443 S 12.500000 P AIR IMS -- -- 13.121179 S

Ray intercept curves (ask the professor for details)

Decomposition of the wavefront on the Zernike polynomials basis, = 5°, = 633 nm *ZERNIKE ANALYSIS Best focus WAVELENGTH 1 Positive angle A is a rotation from the +y axis toward the +x axis. -0.051376: [0] 1 -0.042076: [1] RCOSA -- : [2] RSINA -0.020738: [3] 2R^2 - 1 2.834575: [4] R^2COS2A -- : [5] R^2SIN2A -0.057919: [6] (3R^2 - 2)RCOSA -- : [7] (3R^2 - 2)RSINA 0.026119: [8] 6R^4 - 6R^2 + 1 -0.010290: [9] R^3COS3A -- : [10] R^3SIN3A 0.000781: [11] (4R^2 - 3)R^2COS2A -- : [12] (4R^2 - 3)R^2SIN2A 0.011123: [13] (10R^4 - 12R^2 + 3)RCOSA -- : [14] (10R^4 - 12R^2 + 3)RSINA -0.004587: [15] 20R^6 - 30R^4 + 12R^2 – 1

= 656nm

= 486nm

= 587nm

= 587nm

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FEUILLE RESULTATS - TP N°1 : ABERRATIONS SUR L'AXE

The aim of this sheet is to help you summarize some of the main results and characteristics of the studied lens

DATE : NOMS : ................................................................................................................. BOITE :

LENTILLE SIMPLE

ØAiry = ………………à = 546 nm ; Øtrou maximal = ……………… µm

Dans le mauvais sens :

Gy du viseur= …………………… ; Øtrou (sortie) = …………………… Øtrou (entrée) = ……………………

ØPSF au meilleur foyer =……………………. à = 546 nm

Dans le bon sens :

Gy du viseur= …………………… ; Øtrou(sortie) = ……………………Øtrou(entrée) = ……………………

ØPSF au meilleur foyer =…………………… ………………..à = 546 nm

ØPSF au foyer paraxial = …………………; ØPSF au foyer marginal =…………….

Longueur de la caustique = …………………. mm ………………… mm

Chromatisme axial principal F C F F = …………………. constringence = ………………….

DOUBLET

ØAiry = ………………à = 546 nm ; Øtrou maximal = ……………… µm

Chromatisme axial principal F C F F =………………… mm

Chromatisme axial 2aire =…………………………….. mm

MONTAGE n°2 Dans le mauvais sens :

Gy du viseur= …………………… ;

ØPSF au meilleur foyer =……………………. à = 635 nm ( % d'énergie encerclée)

Dans le bon sens :


ØPSF au meilleur foyer =……………………. à = 635 nm ( % d'énergie encerclée)

OBJECTIF D'AGRANDISSEUR (REF :……………………………………………..)

ØAiry = …………………. à N = ….. et = 635nm ; Øtrou maximal = ……………… µm

Dans le bon sens à pleine ouverture :


ØPSF au meilleur foyer =……………………. à = 635 nm

Jusqu'à quelle ouverture numérique cet objectif est-il limité par la diffraction sur l'axe ? ……..…

Chromatisme axial principal F C F F =………………… mm

Chromatisme axial 2aire =…………………………….. mm

14

RESULTS SHEET – LAB SESSION N°2: OFF-AXIS ABERRATIONS The aim of this sheet is to help you summarize some of the main results and characteristics of the studied lens

DATE : NAMES: ……………………………………………………………………… BOX N° : ..

DOUBLET

ØAiry = ……………… at = 546 nm ; Øpinhole max = …………………

On axis using green light:

gy of the viewer = ……………………… ; Øpinhole = ……………………. mm

ØPSF at the best focus = ……………….. µm

On axis using white light:


At = 5°, in green light :

gy of the viewer = ………………… ; Øpinhole = ……………………. mm

ØPSF au meilleur foyer = ……………….. µm

Dimensions of the focal lines at the tangential focus T= …………………….mm and at the sagittal focus S=…………………….mm

Astigmatism length ST = ………………mm

At which field angle does the image spot start changing significantly?

= ……………° y' = ………. mm

Main aberration: …………………………………………………….

MAGNIFYING OBJECTIVE (REF :……………………………………………..)

ØAiry = …………… at full aperture (N = ) and = 546nm ; Øpinhole max = …………………….

For which aperture numbers is the objective diffraction limited ? .......................................................................................

On axis at full aperture using green light:

Gy of the viewer = ……………………… ; Øpinhole = ……………………. mm


On axis at full aperture using white light:


At which field angle does the image spot start changing significantly?

= ……………° y' = ………. mm

Main aberration: …………………………………………………….

At full field angle (y' = 22 mm), full aperture using green light:

Gy of the viewer = ……………………… ; Øpinhole = ……………………. mm


Main aberration: ……………………………………………………………...

For which aperture numbers is the objective diffraction limited at full field angle ?……..…

15

RESULTS SHEET – LAB SESSION N°3 : ZYGO The aim of this sheet is to help you summarize some of the main results and characteristics of the studied lens

DATE : NAMES : .............................................................................................................. BOX N°:

DOUBLET ØAiry = ……………… at = 633 nm In the best orientation, on axis at the best focus: ∆PV = ……….. ; ∆RMS= ………..

Main Zernike aberrations terms (in units):..................................................................................................................................

......................................................................................................................................................................................................

Main Seidel aberrations terms (in units): .................................................................................................................................

......................................................................................................................................................................................................

Strehl ratio with PSF = ………………; Cut-off frequency at 10% = …………… line pairs/mm

Encircled energy (84%) = …………….…… µm

At = 5°, at the best focus: ∆PV = ………..; ∆RMS= ………..

Main Zernike aberrations terms (in units):...........................................................................................................................

......................................................................................................................................................................................................

Main Seidel aberrations terms (in units): ....................................................................................................................................

......................................................................................................................................................................................................



In the worst orientation, on axis at the best focus: ∆PV = ……….. ; ∆RMS= ………..


......................................................................................................................................................................................................


......................................................................................................................................................................................................



MAGNIFYING LENS ØAiry = ……………… à = 633 nm

On axis, at the best focus at full aperture: ∆PV = ……….. =; ∆RMS= ………..


......................................................................................................................................................................................................


......................................................................................................................................................................................................



At the edge of the field of view, t the best focus at full aperture: ∆PV = ………..; ∆RMS= ………..


......................................................................................................................................................................................................


......................................................................................................................................................................................................



17

LAB SESSION N°1: ON-AXIS ABERRATIONS

THE “POINT SOURCE METHOD”


In this lab session, you will study and compare the on-axis aberrations of three optical systems using the “point source method”, i.e. spherical aberration and chromatism.

The optical systems are the following: a plano-convex singlet, a Clairaut – Mossotti doublet, and a magnifying objective. In particular, you will compare the performance of the singlet and the doublet, which have the same focal length and the same numerical aperture.

Preparatory work: read carefully the principle of the point source method presented in the introductory part of this book and in the text below. Do the calculations labeled in the part §C.

A. Memo

1. Chromatism and glass dispersion

The dispersion of a transparent optical medium is defined as the variation of its refractive index n as a function of the wavelength . This function is usually slowly decreasing in the visible domain. So, it is possible to characterize a glass by the mean value of the index and by the variation n of the index over the visible domain. In fact, instead of n, we use the constringency (also called Abbe number), defined as n 1 n , which is a useful quantity to calculate the chromatic aberrations:

d nd 1

nF nC

where d, F et C denote the following spectral rays:

ray element color wavelength (nm)

D Helium Yellow 587.6

F Hydrogen Blue 486.1

C Hydrogen Red 656.3

18

The available glasses may be represented in a graph (, n) as shown on Figure 1. Remember that the constringency characterizes a dielectric material, not an optical system.

The dispersion of refractive index of materials leads to a variation of the paraxial characteristics of an optical system with the wavelength, called chromatism.

For a singlet, the variation of the focal length is given by: f’ = f’B – f’R = - f’/d.

Figure 4: different glasses in the (, n) plane (courtesy: SCHOTT)

2. Third-order spherical aberration

Spherical aberration appears on axis and remains constant all over the field. It is related to the aperture of the system. Within the 3rd order approximation, using a purely geometrical approach (no diffraction), it can be shown that spherical aberration has the following properties:

- the longitudinal extent of the associated caustic is given by: l FP F a 2

where ’ is the image aperture angle

Crowns

Flints

19

- the radius of the PSF at the paraxial focus is given by: d y l a 3

- the optical path delay (OPD) of the wavefront with respect to a sphere centered on the

paraxial focus is given by: :

For a plano-convex singlet in the infinite-focus conjugation, the coefficient denoted as ‘a’ which characterizes the amplitude of spherical aberration, is given by :

Best orientation (curved side towards infinity):

f

1nn22n2na 2

23

Worst orientation (curved side towards the focus):

f

1n2na 2

2

Note that in order to interpret the PSF that you observe in the image plane of the lenses under test, you need to take diffraction into account.

B. Experimental set-up

You will implement two set-ups during this lab-session:

- The first set-up uses a white light source followed by a monochromator and a microscope viewer that enables visual observations of the Point Spread Function and measurements of the longitudinal chromatism of the lenses under test. The point source is placed far away from the lens under test;

- The second set-up uses a fibered laser diode at =635nm and a CMOS camera that enables recording of the Point Spread Function and comparisons to the diffraction limit. The point source is placed at the object focal point of a collimator.

Set-up n°1 - monochromator and microscope viewer

20

1. Choosing the pinhole

With this set-up, the on-axis aberrations are visually studied for lenses on an infinite-focus conjugation. In principle to measure the effects of the aberrations of the lens on the image spot, the object should be a point source at infinity (that is to say either at a distance of at least ten times the focal length of the lens under test, or in the focal plane of a collimator). In practice, the pinhole should be sufficiently small so that its geometrical image does not limit the resolution of the observation. On the other hand, it should lead to an acceptable amount of light.

2. The monochromator – Uncertainty on the wavelength

The light source is a 70W (white) iodine lamp filtered by a Jobin Yvon monochromator, which is about 5 m away from the optical system under test. This will allow you to measure the spherical aberration for a given wavelength, as well as longitudinal chromatism.

The monochromator is based on a concave holographic grating with 1200 lines/mm and a radius of curvature of 200 mm (2f-2f conjugation) (see Figure 3).

The spectral resolution is 8nm/mm. So for an infinitely small output hole, the spectral width of the light emitted by the monochromator is determined by the diameter of the input hole, i.e. 8nm for a diameter of 1mm.

21

Figure 3: Schematic of the monochromator

3. Choosing the microscope objective

The object numerical aperture of the microscope objective should be larger than the image numerical aperture of the lens under test (in order not to limit the aperture of the lens under test), and the magnification of the microscope objective should be sufficiently large to perform an easy measurement of the PSF.

4. Effect of the finite distance between the point source and the lens under test

The PSF that you observe with the microscope viewer is not rigorously in the focal plane F’ of the lens under test, but rather in the image plane A’ (the conjugate of the point source), because of the finite distance between the point source and the lens (D ~ 5m). In the following, however, you want to determine how the focal length

FHf of the lens under test evolves when you vary the wavelength (to study chromatism). The measured distance AHz is related to f through:

f1

D1

z1

Variations in z and f are thus related through:

Grating

Flat mirrors

Iodine lamp

Input hole Output hole

22

zDf21z

Df1f

2

The measured variations in z should be corrected by the factor

Df21 in order to

evaluate the corresponding variations in the focal length.

C. Preparatory Questions

Simple lens aberrations 1. Calculate the Airy spot diamètre of the lens (λ = 546 nm). 2. Evaluate the primary axial chromatism f of the plano-convex (cf. §A.1). 3. What is the theoretical diameter of the spot due to spherical aberration, in the case

of the singlet, in both orientations and in the plane of the paraxial image? Compare to the diameter of the Airy spot and to the simulations performed with OSLO (cf. Appendix 1).

4. How is the spot diameter modified in the least scatter image plane? 5. What is the length of the 3rd order spherical aberration caustic (cf. §A.2) ?

Doublet 6. Calculate the Airy spot diamètre of the doublet (λ = 546 nm).

Set-up n°1 7. Calculate the magnification between the object plane (output hole of the

monochromator) and the plane of the image of the output hole. 8. What should be the minimum numerical aperture of the microscope objective?

9. Evaluate the correction factor that you should apply to your measurements of A

to deduce the evolution of the focal point F (cf. §B). 10. Evaluate the spectral width of the light source when the diameter of the input hole

of the monochromator is 3mm. 11. Choose the monochromator exit hole diameter to get a geometrical image smaller

than the Airy spot given by the lens or the doublet? 12. For the single lens, choose the monochromator exit hole diameter to get a

geometrical image smaller than the image spot given by the lens at the best focus ?

Magnifier objective The objective will be studied only with the set-up 2. 13. Determine the diameter of the Airy disk at full aperture and at N = 8 and 11 (at λ = 635 nm.

23

14. The single mode fiber has a mode diameter of 4,3μm and a numerical aperture of about 0.10. What will be the diameter of the fiber geometrical image in the image plane of this objective? Compare to the Airy disk diameter. 15. What should be the minimum object numerical aperture of the objective of the microscope?

D. Observations and measurements

1. Characterization of the singlet

The singlet will be studied with the set-up n°1.

The available pinholes have the following diameters: 50 µm, 0.1 µm, 0.2 µm, 0.3 µm, 0.5 µm and 1 mm.

Choose the pinhole diameter adapted to study and measure the size of image spot (the Point Spread Function, PSF) of the singlet.

i) Spherical aberration

ADJUST THE WAVELENGTH OF THE MONOCHROMATOR AT = 546NM.

► Measure the dimensions of the image spot at the best focus for both orientations of the singlet. Identify which orientation is the best, i.e. which orientation minimizes the spherical aberration.

► Compare your measurements with spherical aberration calculations (at 3rd order).

IN THE FOLLOWING, YOU WILL CHARACTERIZE THE SINGLET IN THE BEST ORIENTATION.

► Try to measure the longitudinal positions of the paraxial focus, of the minimum scatter focus, and of the marginal focus. Explain how you recognize them. Draw some sketches to illustrate your observations. Why is it not possible to find accurately marginal position with this method?

Use the diaphragm and the ring diaphragm to measure more precisely these 3 foci

► Evaluate the length of the axial caustic. How does the finite distance between the point source and the lens change your result? Compare your observations to the 3rd order theory.

► Compare your results to the OSLO simulations (see “Appendix 1” in “General presentation of the aberrations lab sessions”).

► Evaluate the uncertainty on your measurements.

24

► Progressively reduce the diameter of the entrance pupil with the iris diaphragm. For which pupil diameter can you consider that the lens is diffraction limited on axis?

Call the assistant professor to cross-check your alignment, observations and measurements.

ii) Chromatism

► Measure the paraxial focus positions for various wavelengths including the F line ( = 486nm) and the C line ( = 656nm) of hydrogen (used in the definition of constringency (Abbe number)).

► Plot FFC versus Compare the correction factor due to the finite distance between the point source and the lens under test with the uncertainty on your focus longitudinal position measurements, and if necessary, apply this correction factor. Add the uncertainty bars on the plot (for positions and wavelengths).

► Evaluate the primary axial chromatism of the singlet, defined as FFC .Compare to the simulation done with OSLO (see “Appendix 1” in “GENERAL PRESENTATION OF THE

ABERRATIONS LAB SESSIONS”).

► Deduce from your measurements the value of the constringency (Abbe number) for the lens material. What glass is the lens made of? Evaluate your uncertainty on the constringency.

2. Characterization of a Clairaut - Mossotti doublet (Thorlabs AC254-150-A1)

You will start studying the doublet with the set-up n°1, and then study it further with the set-up n°2.

► Choose the pinhole diameter to observe and measure accurately the point spread function of this doublet?

► Determine rapidly which orientation of the doublet is the best, by comparing the PSF dimension in both orientations.

IN THE FOLLOWING, USE THE DOUBLET ON AXIS IN THE BEST ORIENTATION.

25

► Show that the chromatism is very low. Evaluate it by choosing three points at both end of the spectrum and in betwin. Deduce the secondary axial chromatism (maximal distance between the foci).

Set-up n°2 - monochromator and microscope viewer

a. The source

In this set-up the source is a fibered laser diode that emits light at = 635nm. The single mode fiber (type SM600) has a mode diameter of 4.3µm and a numerical aperture of ~0.10 at = 635nm. The laser diode is supplied by a stabilized current supply ILX Lightwave that delivers a current up to 40mA.

The collimator

The collimator is a doublet with a focal length f’=500mm it enables the study of the lenses in the ∞F’ configuration.

b. The camera

The CMOS 8bits image detector is 6.4mm x 4.8mm with square 10µm pixels. The camera is controlled by the uEye Demo software which enables the real-time observation, grap and save of the image spot. The acquired image is then analyzed with Matlab (Mesure_PSF function). Make sure that the camera does not saturate and adjust the Exposure Time and/or the current in the laser diode accordingly. A User Manual for the camera and image analysis softwares is available in the room.

The tube length is well suited so that the detector is located in the image plane of the microscope objective and that the magnification is equal to the value indicated on the microscope objective.

26

c. The microscope objective

The microscope objective should be chosen by following the above-mentioned criteria.

d. Alignment procedure

Adjust the orientation of the lens in both directions so as to observe a point spread function with the symmetry of revolution.

► Analyze again the point spread function of the doublet for the best orientation. Observe the encircled energy profile. Compare with the diffraction limited case.

► How does the PSF evolve when you defocus forth and back, around the best focus plane?

► Can you conclude whether the doublet is diffraction limited on axis (or not) for this orientation?

► Turn the doublet around in its worst orientation, and observe again the PSF and the encircled energy profile, in the various focusing planes (paraxial, least scatter, marginal). Evaluate the dimension of the image spot corresponding to an encircled energy of ~84%.

3. Characterization of a magnifying objective

► Analyze the shape of the PSF and measure his diameter for various apertures. Plot the PSF diameter vs N.

► Is the objective corrected of spherical aberration at full aperture? What is the range of aperture numbers for which it is diffraction limited? For which F number the diameter of the PSF is minimal.

27

LAB SESSION N°2: OFF- AXIS ABERRATIONS

THE “POINT SOURCE METHOD”


In this lab session, you will study the off-axis aberrations of two lenses in infinite-focus conjugation. Firstly, you will study the astigmatism and field curvature of a Clairaut – Mossotti doublet, which is virtually corrected of spherical aberration and coma. Then, you will characterize a standard magnifying objective over its full field.

Preparation: read carefully the global introduction to the point source method as well as the text below and to the preliminary calculations (part C).

E. Memo

1. 3rd order coma

Coma is an off-axis aberration that leads to a PSF that resembles a comet (see Erreur ! Source du renvoi introuvable.). Coma is related to the fact that, when the field dimension y (respectively y ) and the aperture angle (respectively ) become large, Abbe’s formula (aplanatic system) sinynsinny is not valid anymore.

In other words, coma corresponds to the fact that the magnification ratio yygy

varies with the aperture angle .

Of course, this happens only for off-axis objects, with height y (or angular height in case the object is at infinite distance). The impact, in the image plane, of a ray issued from an off-axis object with image height y and aperture angle , also depends on the angular position of the ray intercept in the pupil plane, i.e. on the azimuth angle .

Within the 3rd order approximation, the characteristic dimension of a coma PSF is given by:

2sinyb

where ‘b’ denotes the coma parameter (see e.g. “Lectures Notes in Optical Design”, by R. Mercier). The length of the coma PSF is 3see)The associated wavefront defect is given by:

cosyb 3coma

28

Figure 5: Shape of the PSF associated with 3rd order coma

2. Astigmatism and field curvature

Both astigmatism and field curvature are off-axis aberrations; they are associated to the fact that the focal surface is neither unique (astigmatism) nor plane (field curvature).

Field curvature describes how the best focus position varies along the chief ray when the field angle increases: the best image surface gets curved. Astigmatism describes the splitting of the focal surface into two focal surfaces and is due to the fact that revolution symmetry is broken around the mean field ray.

The wavefront departure associated to 3rd order astigmatism is given by:

2cos''y4A 22

ast , where

2CCA TS .

The wavefront departure associated to the field curvature is given by:

22curv ''y

4C

, where

2CCC TS .

29

The tangential rays, i.e. the rays that remain in the incidence plane2 (azimuth angle 0 ), focus in the so called tangential focus, while the sagittal rays, i.e. the rays that

propagate in the perpendicular plane, focus in the so called sagittal focus line (see Figure 6). The spreading of the sagittal rays around the tangential focus leads to a “segment” of light perpendicular to the incidence plane, known as the tangential focal. Similarly, the spreading of the tangential rays around the sagittal focus leads to a “segment” of light parallel to the incidence plane, known as the sagittal focal line. The best focus is in the middle of the tangential and sagittal lines, and resembles a lozenge.

The geometrical analysis shows that the sagittal focus S, the tangential focus T, and the best focus C, are located on spherical surfaces that are equivalent to parabolas, to 2nd order in y (see Figure 7). These focal surfaces are characterized by their curvatures CS, CT, and

C. To 2nd order in y , the s and t defocus distances are given by:

2yCs

2

S

and 2

yCt2

T

Figure 6: Splitting of the focal point into tangential focus T and sagittal focus S, due to off-axis propagation in an optical system and to breaking of axis-symmetry.

2 The incidence plane is defined by the object and the optical axis of the lens.

T S

30

Figure 7: Definition of the tangential (t) and sagittal (s) defocus distances, associated to the tangential (T) and sagittal (S) focal surfaces. The minimum scatter focal surface (C) is in the middle of T and S. The associated curvatures are respectively TC , SC , and C ; the latter is called the field curvature. Here, TC , SC , and C are negative quantities, while the astigmatism curvature 2/CCA TS is positive.

The focal line lengthes, sagittal and tangential, observed in S and T are : with and the diameter of the spot is .

For a thin optical system, one can show that '

1

'

2

fnfC

and

'

1

2 f

CCA TS

.

F. Experimental set-up

1. The light source

The experimental set-up is based on a “point” source, i.e. a pinhole placed at the object focal point of a collimator. The pinhole is illuminated by a LED. You can select the wavelength of the LED by using the rotation switch on the power supply: = 630nm (red), 530nm (green), 470nm (blue); a white LED is also available. Place the selected LED in front of the pinhole by translating the LED bar. Various diameters are available for the pinhole: 12.5µm, 50µm, 100µm, 400µm and 5mm. The collimator is a very good quality lens with a focal length f’ = 500mm. The source ensemble, i.e. the lamp, the hole and the collimating objective, can rotate around a vertical axis. The lens under test is thus illuminated by a point source at infinite distance in a variable direction , which is the field angle(see Figure 8).

The lens under test is centered in a lens holder that you can tilt horizontally and vertically in order to align the optical axis parallel to the bench axis.

31

Figure 8: Experimental set-up

2. Translating the microscope viewer and the collimator

Both the collimator and the microscope viewer are mounted on motorized stages (translation and rotation stages, respectively), which allow you to study the off-aberrations of the lens under test on a wide range of field angles and with a high accuracy. Each stage can be displaced manually. The front panel of the stages controller is shown on Figure 6. The digital displays indicate the horizontal displacement of the translation stage with a resolution of 10µm (bottom display) and the angular displacement of the rotation stage with a resolution of 0.01°.

The motorized stages are very accurate but extremely fragile. Use them with care!

Figure 9: Front panel of the stages controller. “O” brings the stages back to a programmed mechanical reference, while “Z” sets the actual position of the stages to zero (‘000000’). The horizontal arrows enable the translation of the microscope viewer perpendicular to the optical axis, while the vertical arrows enable the rotation of the source ensemble around a vertical axis. Pressing any arrow and “GV” simultaneously enables rapid displacements, either in rotation or in translation.

The counters indicate the relative displacement of the stage compared to the ‘000000’ position.

y ’

f’ = 500 mm

Optical system

collimator

Hole source LED

Optical axis

Input pupil

000000

S

000200

O Z

O Z

GV

Rotation of the collimator

Translation of the viewer

Microscope viewer

32

G. Preliminary calculations

1. Doublet in the best orientation

Q1- Evaluate the astigmatism coefficient A and the field curvature coefficient C of the doublet under test, using its specifications (cf. Appendix 1). Q2- What is the expected dimension of the spot at the best focus for a field angle 5 ?

Q3- Calculate the diameter of the pinhole that enables a measurement of the PSF diameter. Calculate the the pinhole geometrical image diameter in the image focal plane of the lens under test?

Q4- What should be the minimal numerical aperture of the microscope objective?

2. Magnifying objective

Q5- What is the diameter of the Airy pattern at the focal plane of this objective at full aperture?

Q6- Calculate the maximal diameter of the pinhole.

Q7- What should be the minimal numerical aperture of the microscope objective?

H. Observations and measurements

1. Characterization of the doublet in the best orientation

i) On-axis analysis of the image spot

You will want to analyze the point spread functions observed at for different wavelengthes red-green-blue and with white light. In each case, recall the diameter of the Airy pattern at the focal point of the doublet?

► Find the best orientation of the doublet.

► For the best orientation, i.e. the orientation that minimizes the aberrations on axis, is the doublet diffraction limited? Measure his diameter and evaluate the uncertainty.

► Explain the influence of the pinhole geometrical image diameter.

33

Call the professor to cross-check your observations and conclusions.

ii) Off-axis analysis of the spot image: observation and measurement of astigmatism and field curvature

► Measure precisely the longitudinal position of the focal plane.

► Vary the field angle around θ = 0° by turning the collimator progressively. Always make sure that the lens is fully illuminated.

Is this doublet well corrected of coma?

► Vary the field angle around = 0° and measure the defocus distances of the sagittal3, tangential4 and least scatter planes with respect to the paraxial plane, for

100 by a step of 2°, and also for = -5° and for = -10° . Evaluate the uncertainty on your measurements.

► Plot the positions with respect to the paraxial focal plane of the sagittal focus S, tangential focus T, and the least scatter focus C with respect to the paraxial focal plane, as a function of the field. You may plot s, t and c as a function of θ (use Excel). Indicate which focal surface is tangential or sagittal on your graph.

► What is the sign of the field curvature for this doublet?

Are the astigmatism and the field curvature of this doublet well described within the 3rd order theory? Justify your answer.

Compare your measurements for curvatures CS and CT to your theoretical predictions (cf. A.2)

► Rotate the lens under test by exactly 5° and measure the dimension of the focal lines and of the minimum scatter circle. Evaluate the uncertainty on these measurements.

Compare your measures to your theoretical estimation of the spot size in the various focal planes, and to the simulation of the PSF performed by the ZYGO in LAB SESSION N°3?

REPEAT YOUR MEASUREMENTS WITH WHITE LIGHT.

Call the professor to cross-check your results.

3 Focal line oriented towards the optical axis of the lens under test. 4 Focal line parallel to the rotation axis of the collimator.

34

► Try to evaluate experimentally for which field angle the aberrations start to significantly degrade the PSF of this lens (i.e. the field angle for which you can notice the smallest astigmatism).

► Calculate the corresponding OPD of aberration for this angle.

2. Characterization of the magnifying objective

This objective was designed to image a photography film 24mm x 36mm onto paper. The grain of the photographic film is approximately 20 µm.

Analyze the performance of the objective on axis and off axis as in H.1, in monochromatic light and in white light and at several apertures, for an object point at infinity.

► In particular, evaluate the dimension of the image spot on axis versus the numerical aperture.

For which aperture numbers is the objective diffraction limited? Compare your measurement with the values that you simulated from a wavefront analysis (cf. LAB

SESSION N°3).

► How do the dimension and shape of the PSF evolve with field angle ? Plot the size of the image spot versus the field angle , over the full useful field.

What are the main aberrations? Does this objective have field curvature? Astigmatism?

At which field angle does the image spot start changing significantly? What is the main aberration?

How does the quality of the PSF evolve when you vary the aperture of the objective, at full field angle (y’ = 22mm)?

Above which aperture number is the objective diffraction limited, at full field?

► Compare your results with the simulations that you obtained with the ZYGO analysis (cf. LAB SESSION N°3).

► Suggest a method to evaluate the lateral chromatism at full field. Try to measure it if you have time.

Suggest a method to measure the distortion of this objective.

Call the professor to cross-check your observations.

35

LAB SESSION N°3: ZYGO WAVEFRONT ANALYSIS BY

INTERFEROMETRIC MEASUREMENTS


The ZYGO belongs to the family of Fizeau interferometers. It allows remote measurements of surface defects and wavefronts transmitted by optical systems. The light source is a HeNe laser ( = 632.8 nm) with a coherence length larger than 100 m. The pupil of the ZYGO is 102 mm. The CCD camera used for the acquisition of the interferogram is a 320 × 240 pixels chip, which is enough to resolve 90 fringes over the pupil of the instrument. The “Phase Shift” method is explained in Appendix 1. It allows wavefront measurements with accuracy better than over the entire pupil.

Preparation: read carefully the global introduction about wavefront measurements and the text below. Do the preliminary calculations in section C.

!!!Very Important!!!

The reference flat of the ZYGO is excellent (a few /100 RMS) but it is extremely fragile and expensive. Be very careful! Never touch it! Always stay away from it and always put the plastic protection back on the reference flat after use.

36

A. Interferometric wavefront measurements

1. Interferogram

Figure 10 : Principle of the wavefront interferometric measurement

A perfect lens transforms a plane wavefront into a spherical one: this is known as rigorous stigmatism. If this wave converges at the center of curvature of a spherical mirror (see Figure 10), the reflected wavefront is again transformed into a plane wavefront. If the adjustment is perfect and if the objective under test is perfectly stigmatic, the Optical Path Delay (OPD) is a constant over the surface of the pupil and the interferogram is uniform, i.e. there is no fringe visible in the interferogram.

For a real lens with aberrations, we observe fringes corresponding to the optical path difference between the real wavefront and the ideal spherical wavefront. This fringe pattern contains information about the geometrical aberrations of the optical system (excluding the chromatic aberrations).

Because the incoming wavefront travels twice through the optical system under test, it experiences the defects of the system twice. Thus, from one white fringe of the interferogram to the next one, the OPD, which is defined as the defect of the wavefront accumulated after one way propagation only, varies by λ/2

The only limitation is the pupil size of the system. The system under test must always be the pupil of the setup. If a spherical reference mirror is used to test a system, the aperture of this spherical reference mirror must be larger than the aperture of the

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tested optical system. If one uses a convex mirror the radius of curvature must be smaller than the back focal length of the tested system.

2. Fringes visibility

The HeNe laser used in the ZYGO is stabilized and single mode, and has a very long coherence length. Thus, the visibility of the fringes is not limited by spatial nor temporal coherence. The visibility may be low only if the intensities of the two interfering waves are very different. The reflection factor of the reference flat is 4% (glass/air interface). The spherical mirror should not be aluminized. The contrast of the fringes is thus excellent. The main limitation of the ZYGO is when the quality of the objective is too poor. In that case, there are too many fringes and the video acquisition system cannot resolve them.

3. Interferogram analysis

The shape of the interferogram geometry sometimes allows you to determine what the main aberration of the system under test is, especially when it is limited to the 3rd order. The 3rd order aberrations have characteristic interferograms that are shown in the Appendix 1. The amplitudes of the aberrations are obtained by decomposing the wavefront over the Zernike polynomials basis (see Table 1 in “GENERAL

PRESENTATION OF ABERRATIONS LAB SESSIONS”). The Zernike coefficients are calculated by the ZYGO software by least squares fitting the measured wavefront. The software calculates with a 2D FFT the PSF and the MTF by using this decomposition and by Fourier Transform, and evaluates the Strehl ratio.

4. Remarks

The ZYGO cannot distinguish the defects due to the optical system under test from those due to the reference mirror. This is due to the principle of the measurement method. However, the contributions of the flat reference (/100 RMS) and of the spherical mirror (/20 RMS) are negligible, in as much the defects of the system under test are larger than /20 RMS. Moreover, the spherical mirror should operate in a configuration where it is stigmatic, which is the case when the beam focuses near the center of curvature. However, if the system under test is of poor quality, this is no longer true and the defects of the mirror are not negligible: then in terms of ray propagation, the rays intercept the plane of the center of curvature of the spherical mirror with a large dispersion, and the spherical mirror cannot be considered as stigmatic any more.

Using a reference sphere (instead of a reference plane) allows you to characterize the performance of an optical system for conjugations other than infinite-focus.

The wavefront measurement is done by comparison to a reference flat; the result of the measurement does not depend on the numerical aperture of the lens under test. However, to perform simulations of the PSF and of the MTF correctly, the

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software requires that you enter the f-number (the aperture number) of the lens, since transverse dimensions and spatial frequencies need to be calculated.

How to use ZYGO – GPI

Launch the MetroPro software under the ‘Eleve’ session and click on the “TP2A-Aberrations.app” application.

5. Adjustments of the optical system under test

The optical set-up is easy and quick to align.

► In ‘Align Mode’ (ALIGN), make sure that the reference flat is well aligned: the retro-reflection should hit the target, on the video monitor.

► The optical system under test should be carefully centered on the pupil of the ZYGO and on the axis of the ZYGO. Align it by auto-collimation, i.e. by using a small aluminized flat mirror and by looking at the retro-reflection on the video monitor.

► Choose a spherical mirror that is well adapted to the back focal length of the lens under test.

► Adjust the position of the optical system under test so that it focuses the incoming laser beam onto the center of curvature of the spherical mirror (not on the vertex of the mirror!). This is done by auto-collimation again.

► In ‘View Mode’ (VIEW), adjust finely the position of the mirror to obtain a flat interferogram, or to minimize the number of fringes.

► Adjust the CAM in order to conjugate the pupil of the system under test with the measurement plane.

► Adjust the ZOOM in order to obtain a large image of the pupil on the video monitor, without clipping.

2. Adjustments of the optical system under test

► Enter the f-number of the lens under test (f-number).

► Launch the wavefront measurement by clicking on MESURE. The software uses a phase-shifting method: it reconstructs the wavefront from the acquisition of 7 interferograms, for 7 positions of the flat reference.

► By default, the software defines a circular mask that covers 98% of the pupil (Masque Auto: Yes). You can visualize the mask in the ‘Masque” window

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(clicking on MASQUES) or on the video screen. You can also define it yourself by un-clicking on Masque Auto: Yes, and then select ‘Acq’ (acquisition mask) and define your mask (“Define’).

► The Topographie du front d’onde window displays the wavefront and the statistics of the wavefront (PV and RMS values). Some defects may be subtracted from the measured wavefront before display, such as the Piston (PST), the Tilt (TLT), the sphere that best fits the wavefront (PWR), and 3rd order aberrations such as spherical aberration (SA3), coma (CMA) and astigmatism (AST). The substraction of one of these terms may be observed without launching another wavefront measurement by clicking ANALYSE.

► The Profil du Front d’Onde window allows you to plot a profile of the wavefront along any cross-section.

► The Décomposition Modale window displays the first 36 Zernike polynomial coefficients and the Seidel coefficients for 3rd order aberrations, in units of . The quality of the fit is also displayed (rms): it corresponds to the RMS value of the residuals. See Table 1 in the “GENERAL PRESENTATION OF THE ABERRATIONS LAB

SESSIONS”.

The Seidel coefficients are calculated from the 3rd order Zernike coefficients. They are displayed together with the TILT and FOCUS, which are calculated with respect to the paraxial focus. These coefficients correspond to the OPD on the edge of the pupil, in units of , and are calculated by using the following relationships:

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Amplitude Angle

TILT 22

21 CC 12 CCarctan

FOCUS 2C3 6C8

ASTIGMATISM5 25

24 CC2 452

1 CCarctan

COMA 27

26 CC3 67 CCarctan

SPHERICAL 8C6

It is noteworthy that these relationships are valid if the lens exhibits 3rd –order aberrations only; it is thus absolutely necessary to check whether high-order aberrations are negligible before using them.

1.

6. Diffraction analysis

The ZYGO software can simulate the PSF and the MTF from the measured wavefront, by Fourier Transform.

► You need to enter the ‘F-number’ in order to obtain relevant values for the PSF and the MTF.

► The Réponse Percussionnelle window calculates the PSF of the optical system and displays the Strehl ratio, i.e. the maximum intensity normalized to the maximum intensity reached by a diffraction limited system. It takes into account the subtracted defects (PST, TLT, PWR…). Encircled Energy displays the integrated power versus the radial coordinate. Cursors (right click >> Show Controller) give access to values of the encircled energy plot. You may change the resolution of the calculation of the PSF by changing the point size (‘taille du point’). For instance, HiRes will let you evaluate the size of the PSF with a better accuracy for systems with little aberration.

5 The Seidel polynomial associated to ASTIGMATISM is of the form u2 cos2 , which differs slightly from the standard form introduced in the Optical Design course at IOGS u2 cos2 ). The

u2 cos2 form follows the Born & Wolf definition. The astigmatism coefficient calculated by ZyMOD/PC thus corresponds to the Peak-to-Valley amplitude of the OPD associated to astigmatism, and not to the amplitude of the OPD on the edge of the pupil (as is true for the other Seidel coefficients).

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► The Calcul de la FTM window displays the MTF, calculated by Fourier Transform of the PSF, along 4 directions. The cut-off frequency 1/N is calculated from the f-number. The MTF profiles are displayed with the following color code: 0° (dark blue lozenges), 90° (green triangles), +45° (light blue squares), and -45° (red).

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B. Preliminary calculations

The specifications of the lenses are gathered p.11.

1. Doublet in the best orientation Q1- What is the doublet f-number? What is its image numerical aperture? Q2- What is the diameter of the image spot if the doublet is diffraction limited? Q3- What is the value of the doublet MTF cut-off frequency at 0% assuming that it is diffraction limited? And what would be the spatial frequencies for MTF values of 10% and 50 % ?

2. Magnifying objective The magnifying objective will be studied at N=5.6 on axis and at the edge of the field of view (y’= 22 mm). Q4- What would be the diameter of the image spot if the objective was diffraction limited? Q5- What is the cut-off frequency of the MTF of the objective assuming it is diffraction limited ? and for 10% and 50% MTF values ? How these values are modified for increasing N values ? Q6- What is the maximal useful field angle of this objective?

B. Measurements

1. Doublet in the best orientation

i) WavefrontmeasurementONAXIS

► Place the doublet in the best orientation on axis and adjust it in order to obtain a flat interferogram.Enter the f-number into the software.

What is the variation of the wavefront error between two consecutive fringes?

► Evaluate the maximum OPD by observing the interferogram on the video monitor. Then, launch the wavefront measurement by the ZYGO. Make sure that the measurement is performed at the best focus.

How do you make sure that the measurement is performed at the best focus?

Why should you subtract the tilt and power terms (TLT and PWR)?

What is the value of the maximum OPD measured by the ZYGO (PV and RMS)? Does it fulfill the Maréchal criterion?

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What are the main aberration terms here? What is their order? Compare to the simulation (cf. Appendix 2).

How can you evaluate the precision of your measure?


► Calculate the PSF. Analyze the shape and the dimension of the PSF. Plot the encircled energy and compare your result with the encircled energy diagram of an Airy function.

What is the Strehl ratio of the PSF. Is the doublet diffraction limited on axis?

Is this result consistent with the measurement of the PSF performed by the point source method (see LAB SESSION N°1)?

► Calculate the MTF and compare your result with the MTF of a diffraction limited system in the same conditions.

? What is the spatial frequency that corresponds to a visibility of 10% and 50% for a sinusoidal target?

ii) WavefrontmeasurementOFFAXIS

► Turn the objective around the vertical axis to clearly observe astigmatism (field angle 5 ). Measure precisely the rotation angle. Observe the different interferograms when the center of curvature of the spherical mirror is respectively at the sagittal focus, the best focus and the tangential focus. Perform the following characterization for each position:

► Measure the wavefront and explain the shape of your result.

Deduce from your 3 measurements the amplitude of the astigmatism and the associated uncertainty.

► Calculate the PSF.

Is the doublet diffraction limited in this configuration? Compare the PSF to the Airy function and to the PSF that you obtained on axis.

Calculate the PSF. What is the Strehl ratio? What is the radius of a disk that encircles 84% of the energy?

Compare your result to the direct measurements of the PSF that you performed by the point source method (see LAB SESSION N°2).

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► Launch the MFT calculation. Determine the spatial frequency at 50 and 10% visibility. Compare your results to the results that you obtained on axis and to the MTF of a perfect system. In particular, observe how the shape and the symmetries of the MTF are related to the shape and symmetries of the PSF.

2. Doublet in the worst orientation

► Place the doublet in the worst orientation and adjust it so that it operates on axis: the interferogram should have the symmetry of revolution.

► Observe the evolution of the interferogram for different positions of the center of curvature of the spherical mirror: paraxial focus, best focus, and marginal focus.

How do you recognize which focus is which?


i) Observationatthebestfocus

► Place the center of curvature of the spherical mirror at the best focus and answer the same questions as previously.

► Evaluate the amplitude of the OPD on the edge of the pupil by observing the interferogram on the video monitor. Then, launch the wavefront measurement.

Evaluate the PV and RMS OPD measured by the ZYGO? Compare with your observation on the video monitor. Is the doublet diffraction limited in this configuration?

► Analyze the wavefront decomposition on the Zernike basis and on the Seidel basis. Verify by yourselves the calculations of the Seidel coefficients done by the ZYGO.

What is the main aberration of the doublet in this configuration?

► Calculate the PSF. Evaluate the Strehl ratio in this configuration and the radius of the disc that encircles 84% of the energy.

Compare the calculation of the PSF performed by the ZYGO at the best focus to the direct observation performed in LAB SESSION N°1 by the point source method.

Compare to the result obtained for the doublet in the best orientation.

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► Calculate the MTF of the doublet. Compare the result to the MTF of a diffraction limited lens, et determine the cut-off frequency at 10%.

Are the shape and the symmetry of the MTF related to the shape and the symmetry of the PSF?

ii) Observationintheparaxialfocus

► Place the center of curvature of the spherical mirror on the paraxial focus of the doublet by translating the mirror longitudinally.

What is the shape of the interferogram at the paraxial focus?

► Evaluate the amplitude of the OPD on the edge of the pupil by observing the interferogram on the video monitor. Then, launch the wavefront measurement. Check that the spherical term (PWR) is not removed for this measurement.

Why is it important to keep the spherical term here?

Evaluate the PV and RMS OPD measured by the ZYGO? Compare with your observation on the video monitor. Is the result compatible with the measurement done at the best focus?

Is the name “best focus” relevant to describe the previous configuration? How is it defined?

► Analyze the wavefront decomposition on the Zernike basis and on the Seidel basis. By comparing the measurements done at the various foci, evaluate the uncertainty on the coefficients of the main aberrations.

► Simulate the PSF. Observe the shape and dimension of the PSF.

Is this calculation valid?

3. Study of a magnifying objective

► Enter the f-number of the objective into the software

► Study the aberrations of this objective on axis, at full aperture, and at the edge of the field of view (y’= 22 mm).

► Analyze the wavefront decomposition on the Zernike basis and the amplitude the OPD.

What are the main aberrations of this lens?

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Does it fulfill the Maréchal criterion on axis? At full field?

What are the uncertainties of these measures?

► Evaluate the dimension of the PSF on axis and at full field.

► Evaluate the MTF of this objective on axis and at the edge of the field of view. What is the spatial frequency at 10% visibility?

What do you think are the most relevant parameters to evaluate the performance of this lens?

Compare your results to those obtained by the point source method (see LAB SESSIONS N°1 & 2).

Comment on your results in the perspective of photography applications. What are your conclusions?

What other measurements are absolutely necessary to fully characterize the performance of this objective, in the perspective of photography applications?

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APPENDIX 3: THE PHASE SHIFT METHOD

The ZYGO interferometer is a two wave Fizeau interferometer. The reference flat is a very good quality plane surface (RMS flatness of a few /100). It can be moved by a piezoelectric device over one or two . In the case of a two wave interferometer, the intensity at any point of the interferogram varies as a sinusoid when the reference flat is translated linearly. All phase-shift algorithms are based on this property.

The images of the interferogram of the system under test are digitalized by a video acquisition card and displayed on a CCD camera. For instance, the picture below displays the interferogram of a system that has astigmatism.

The reference plane moves by /10 between two images.

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The figure below shows the sinusoidal evolution (with period /2) of the intensity recorded by three pixels, denoted as A, B, and C, which correspond to three different points indicated on the interferogram during the displacement of the reference flat.

The relative phases of these sinusoidal functions are directly related to the shape of the wavefront that is reflected by the surface or transmitted by the optical system under test. Numerous Phase-Shifting algorithms try to determine as accurately as possible the phase between sinusoidal functions. Here, ZYGO takes 6 images (about 220 x 180 pixels). One of the main advantages of the phase-shifting method is the spatial resolution: the wavefront is measured on each pixel (200 x 180 = 36000 pixels). The displacement of the reference flat is calibrated in factory: it is translated by /8 precisely (the corresponding

phase shift is /2 between two interferograms).

With five images, the phase shift angle may be calculated by the formula:

1 5 1,

4 2

1 cos ( )

2i j

I I

I I

where I1, I2, I3, I4, I5 are the grey levels of any pixel in each of the five images.

49

Before each calculation of the wavefront, the ZYGO software calculates the i,j phases and the mean value of these phases, and verifies that the mean value is 90°+/-1°.

Then, the Hariharan algorithm with 5 images is probably used by the ZYGO software. The surface under test is given by:

)2

)(2arctan(

4),(

153

42

III

IIyxh

The phase determination is given modulo 2π radians. So the phase has to be unwrapped (removing 2π ambiguities). This algorithm is based on the hypothesis that the surface has no 2π, or more, steps (the surface is supposed to be smooth enough). This is a drawback of any interferometric methods (except in white light interferometers).

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APPENDIX 4 :

EXAMPLES OF INTERFEROGRAMS AND WAVEFRONT DEFECTS DUE

TO 3RD ORDER ABERRATIONS.

1. Defocus (FOCUS = 2)

Here, u denotes the aperture variable, normalized to the edge of the pupil.

2. Spherical aberration (SPHERICAL = 2)

Paraxial focus Best focus Marginal focus

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3. Coma (COMA = 4)

Paraxial focus With a tilt (best focus)

Cross sections of the wavefront in the tangential plane

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4. Astigmatism (ASTIGMATISM = 6)

Center of curvature of the spherical mirror at the best focus

Center of curvature of the spherical mirror at the sagittal / tangential focus

53

LAB SESSION N°4: HASO SHACK-HARTMANN WAVEFRONT

ANALYSIS


Shack-Hartmann wavefront sensors are widely used in various fields of optics, such as adaptive optics, optical bench alignments, etc… They allow accurate measurements of the geometrical aberrations of optical systems. The wavefront sensor called HASO (the French acronym for Hartmann Wavefront Analyzer) was developed by Imagine Optics, a company founded by alumni of Institut d’Optique Graduate School.

Preparation: THE LAB REPORT OF THIS LAB SESSION SHOULD BE GIVEN IN

AT THE END OF THE SESSION. It is thus necessary that you prepare this lab-session ahead of time, by reading carefully the subject and answering the preparation questions (§C). In particular, you should read carefully the global introduction about wavefront measurements.

A. Description of the HASO ................................................................. 54

1. Principle ......................................................................................... 54 2. From local slopes to wavefront measurements .............................. 55 3. Evaluation of the image spot: PSF and spot diagram ..................... 56 4. Wavefront measurements with the HASO ..................................... 57 5. Adjustments of the HASO .............................................................. 60 6. Using the image analysis tools ....................................................... 62

B. Experimental set-up .......................................................................... 63 C. Preparation questions ........................................................................ 65 D. Measurements .................................................................................. 66

1. Direct analysis of the PSF on axis .................................................. 66 2. Study of the lens on axis at full aperture with the HASO .............. 67 3. Fine study of the PSF through direct measurement ........................ 68 4. Study of the magnifying objective off axis at full aperture ............ 69

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A. Description of the HASO

a. Principle The HASO 32 is an array of 32 32 micro-lenses associated with a 5 mm 5 mm CCD camera. The CCD matrix is placed in the focal plane of the micro-lenses. These lenses are 160 µm-square lenses with a 6 mm focal length (i.e. the aperture number is N = 37.5). The CCD matrix has 512 512 square pixels. The pixel size is 10 µm ×10 µm , so that the micro-lenses array has exactly the same surface as the CCD chip: each micro-lens is associated to an area of 16 16 adjacent pixels.

The HASO is illuminated by the wavefront to be analyzed, which is sampled by the micro-lenses array. Each micro-lens forms an image spot in the CCD plane. If the wavefront is plane and perpendicular to the lenses axis, each spot will be on the optical axis of the associated micro-lens. On the contrary, if the wavefront is tilted with respect to the micro-lenses array or is not a plane, the positions of the spots will depend on the local slope values of the wavefront sampled by each micro-lens (see Figure 11).

16 16 pixels

PSF of the

Optical axis of the micro-

160 m

160 m

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Figure 11 : Principle of the wavefront measurement

The HASO software measures accurately the mean displacement of each spot nn y,x with respect to the nth micro-lens axis. The mean displacements are related to the local slope of the wavefront in each direction by f/xnx and f/yny . The software is able to

associate each spot with the corresponding lens for displacements larger than the micro-lens size. This feature is based on the shape comparison of each spot to recorded reference spots for each of the 1024 micro-lenses. It increases the dynamics range of the HASO noticeably, and is launched by the TRIMMER button. The accuracy guaranteed by the manufacturer on the wavefront measurement is /100 RMS.

b. From local slopes to wavefront measurements

The HASO software displays the values of the local slopes measured right after each micro-lens. The wavefront is deduced from the local slopes either by direct integration (zonal method) or by decomposition of the whole set of measured slopes over the derivatives of a basis of polynomials that are suited to the pupil geometry (modal method): we will use the latter in the following. Two polynomial bases are available: the Legendre polynomials are suited to rectangular pupils, while the Zernike polynomials are suited to circular pupils. Each polynomial corresponds to a given defect of the wavefront. This allows a fine analysis of the

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wavefront quality as well as the evaluation of the budget of aberrations. The wavefront can be decomposed upon the first 64 Zernike polynomials6 by least squares fitting the set of measured slopes to the derivatives of the Zernike polynomials.

Important note: whatever the method used to analyze the wavefront, the software fits the measured wavefront to a sphere (often called the reference sphere). Hence, it calculates the mean radius of curvature of the wavefront in the plane of the micro-lenses array. Knowing the pupil diameter and the radius of this sphere, the HASO software deduces the numerical aperture of the optical system under test as well as the best focus position. The “best focus” is defined as the center of the sphere that best fits the wavefront, i.e. the sphere that minimizes the standard deviation RMS of the optical path delays with respect to the measured wavefront. Hence, the reference sphere used by the HASO software is automatically centered on the best focus. c. Evaluation of the image spot: PSF and spot diagram Starting from the wavefront measured in the plane of the micro-lenses array, the HASO software can calculate the PSF in the best focus plane by implementing a two-dimensional Fast Fourier Transform (2D FFT). This calculation takes into account the local illumination measured after each micro-lens. The ratio of the

6 This decomposition is possible if at least 64 microlenses are illuminated. The software proposes two normalization modes for the Zernike polynomials; the RMS normalization is used here, in agreement with the definitions given in class.

R

’Sphere

Measured wavefront

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maximum of the real PSF (i.e. with a distorted wavefront) and the maximum of the diffraction limited PSF (i.e. with a perfectly spherical wavefront) yields the Strehl ratio of the optical system under test in the measurement configuration. The HASO software can also calculate the Modulation Transfer Function (MTF), again starting from the measured wavefront. Starting from the set of measured slopes, and using the fact that optical rays are perpendicular to the wavefront, the HASO software can also plot the Spot diagram, which is the intersection of the optical rays with the best focus plane. d. Wavefront measurements with the HASO

Wavelengthofthelightsourceandexposuretime

The HASO performs a geometrical measurement of the wavefront, which is thus independent of the wavelength. Besides, the measurement is possible only in the spectral range of the CCD chip, i.e. 350 – 1100nm. If the light source is polychromatic, the measurement is the average of the monochromatic measurements over the spectrum of the light source. In order to make the signal sufficiently large behind each micro-lens, the integration time of the CCD sensor must be adapted to the illumination of the optical system under test. Ideally, the CCD should operate near saturation, i.e. above 90% of the saturation level. The “Exposure duration” can be adjusted manually or automatically in the Configuration window by using the Auto exposure option. If “Auto exposure” is selected, make sure that the micro-lenses array is sufficiently illuminated to allow a measurement within typically 300ms.

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Startinganewacquisition

Click on the green arrow to start a new acquisition. The measurement pupil (in white) is automatically determined and corresponds to the micro-lenses that are sufficiently illuminated to perform a measurement7. In case of a modal decomposition of the wavefront over the Zernike polynomials, a blue circle indicates the pupil over which the wavefront is analyzed.

Then, it is necessary to associate each image spot with the corresponding micro-lens. This operation is done by clicking on

the TRIMMER button and takes approximately 1s. This enables the software to track the displacements of the image spots on the CCD with respect to their associated micro-lenses. The algorithm used for this tracking is based on a displacement probability that sometimes fails when the number of correctly 7 Fluctuations of the diameter of this pupil may occur if the intensity of the light source fluctuates or if the marginal microlenses are not sufficiently illuminated, in which case you should improve the illumination conditions or increase the exposure time. A more radical alternative consists in deleting the marginal microlenses when you define the pupil and measure the wavefront (window PENTES>> Shut off boundaries =1). The main drawback of this technique is that the performance of the optical system under test is not evaluated at full aperture! Use this option only if all other solutions have failed …

Acquisition

Measuring the local slopes

Results

Configuration

Starting the acquisition

Image spots display

Measurement pupil

Local slopes display

Referencing the image spots with respect to the micro-lenses (Standard mode / Trimmer mode)

Acquisition

Configuration

Illumination level of the camera

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illuminated micro-lenses is too small. In this case the tilt measurement fluctuates. It is possible to cancel this spot tracking by clicking on the NEW LENS button.

Slopesmeasurementsandresultsanalysis

Starting from the measured set of slopes, the HASO software performs various operations to analyze the wavefront:

o Slopes measurements: the HASO software maps out the displacement vectors for each micro-lens, starting from the micro-lens axis and ending on the image spot center. The length of each vector is proportional to the local slope of the wavefront. In order to observe the defects of the wavefront due to the aberrations, you should subtract the tilt and focus. This window also evaluates the number of illuminated sub-pupils (i.e. the micro-lenses) and allows you to re-define the measurement pupil.

o Beam parameters: this window displays the beam parameters such as the tilt in two orthogonal directions, the mean radius of curvature, the position of the best focus with respect to the micro-lenses array, and the astigmatism of the beam.

o Spot diagram: this window displays the spot diagram near the best focus plane (defocus=0). The spots correspond to the intersections of the rays with the plane of observation, with a

Slopes measurements

Spot Beam t

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color scale associated to the areas in the pupil where the rays come from (marginal rays / paraxial rays). Adjust the zoom cursor if the spots are too dispersed on the spot diagram (Window size).

The following buttons are links to the main quantities that characterize optical systems:

o Intensity: displays the measured intensity that illuminates each micro-lens.

o Wavefront: displays the wavefront and quantities such as the Peak-Valley amplitude of the wavefront as well as the standard deviation of the wavefront fluctuations.

o Polynomials coefficients: displays in real time the calculated coefficients of the wavefront decomposition on the Zernike basis (only in “modal reconstruction” of the wavefront). The coefficients are displayed in µm. This window also displays the diameter of the pupil and the measured numerical aperture of the wavefront that illuminates the micro-lenses array.

o Expert: displays the PSF, which is calculated by 2D Fourier Transform of the wavefront defects, the Strehl ratio, and the MTF, which is obtained by 2D Fourier Transform of the PSF. This calculation can also take into account the intensity distribution on the pupil (recommended) by clicking on Intensity >> Measured.

e. Adjustments of the HASO

Measured intensity

Wavefron MTF and PSF calculations

Polynomials coefficients

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Adjusting the HASO is very simple. First, the optical system under test should be adjusted on the axis of the collimator. Then, the HASO should be placed slightly behind the focal point of the optical system under test.

► Start the acquisition and adjust the time exposure of the camera if necessary (see 4.i).

► Adjust the position of the HASO camera transversally and longitudinally so that the incoming beam is well centered on the CCD and fully contained in the array of micro-lenses. Check that it is the case by clicking on the alignment button and by observing the image spots on the CCD.

► Adjust the HASO camera tip-tilt wise so that it is perpendicular to the incoming beam. To do this, an alignment pinhole is available on the front of the HASO camera. Insert it, click on the

button to operate in “alignment mode”, and adjust the camera tip-tilt wise until the red dot is inside the green circle. This procedure ensures that the spot images are correctly associated to their corresponding micro-lenses. This should result in a better evaluation of the wavefront. This adjustment is not critical, though, since the HASO can measure tilts up to 3 .

► Click on the NEW LENS button or on the TRIMMER button to initialize the association of the image spots with the micro-lenses (see 4.ii). It is necessary to re-initialize this association after every major change in the alignment of the system, e.g. the CCD or the lens under test.

► Observe the Zernike coefficients (Modal coefficients ). Check that the numerical aperture of the lens under test, as measured by the HASO, is correct.

Provided the incoming beam is fully contained in the micro-lenses array, the Zernike coefficients associated to field aberrations, e.g.

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coma and astigmatism, are very convenient to adjust finely the orientation of the lens under test parallel to the collimator axis. Adjust finely the tilts of the lens under test to make the optical axis parallel to the collimator axis, until these coefficients are minimized.

f. Using the image analysis tools

► The µEye Cockpit software let you see the point spread function (Live Video) as imaged by the microscope objective on the camera. To measure its typical dimensions, you need to define the measurement unity corresponding to one CMOS pixel, which depends on the magnifying ratio of the objective: Menu Dessin ≫ Mesure ≫ Définir une unité de mesure. To observe PSF profiles, choose Menu Vue ≫ Vue ligne horizontale or verticale. The tool "Mesurer la distance" in the sidebar draws a line and prints its length in the chosen measure units.

► The Matlab script Mesure_PSF allows you to analyze the image; its operating manual is available in the lab room. It is specifically suited to evaluate the percentage of energy inside a circle of given radius (encircled energy) in comparison to the diffraction limit. Be careful of recording a non-saturated image of the PSF, and defining a wide zone around it.

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B. Experimental set-up

The light source used in this experiment is a laser diode ( = 635nm); the point source is the output of a single mode fiber with a mode diameter of 4.3µm at 635nm and a numerical aperture of 0.10. The laser diode is powered by a stabilized current supply, which you can adjust to control the luminosity.

Figure 12 : Experimental set-up

The optical system under test is a f' = 150 mm doublet, with a diameter of 25 mm (Thorlabs AC245-150-A) designed to operate for an infinite conjugation in the visible. It is used with an iris diaphragm. The doublet is maintained in a mechanical support that is adjustable in orientation. Here, it will be characterized in a configuration where the transverse magnification ratio is gy = -1/3.

A removable mirror as well as a microscope viewer is available to observe directly the PSF of the system under test on the CMOS camera, and compare the measurements performed by the HASO to your observations. The choice criterions for the microscope objective are similar to those explained in “THE POINT SOURCE

METHOD” in “GLOBAL INTRODUCTION TO ABERRATIONS LAB

SESSIONS”. The MatLab script Mesure_PSF allows you to further analyze the PSF. Remark: like in the ZYGO set-up, one should in principle insert an extra lens in the focal plane of the optical system under test so as to conjugate

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the pupil of the optical system under test and the micro-lenses array of the HASO.

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C. Preparatory questions

The following questions need to be answered before the lab sessions.

1. Principle and analysis of the HASO instrument

a. What are the shape and the size of the PSF of a micro-lens? Compare to the size of the pixels. Do you believe that the spatial sampling of the diffraction spot by the pixels is well suited to the calculation of the position of the barycenter?

b. The HASO cannot measure a wavefront with a radius of curvature smaller than 25 mm. What is the corresponding maximal numerical aperture which can be measured?

c. According to the manufacturer, the maximum measurable wavefront slope is 3°: what is the corresponding spot displacement with respect to the micro-lenses axis? Compare this value to a microlens size and comment.

2. Operating conditions of the doublet The doublet should work for a conjugation ensuring a -1/3 magnifying ratio, and will be evaluated for a field angle between 0° and 2°. In the following we shall assume that it behaves as a thin lens.

a. What should be the distance between the point source and the lens under test in order to ensure the wanted magnifying ratio?

b. Estimate the numerical apertures (in object and image spaces), and compare them with the NA of the source fiber and the maximum NA measured by the HASO respectively.

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c. Calculate the Airy spot diameter in the image plane. Compare the image of the fiber core diameter to this value.

d. What is the lateral position of the image corresponding to the 2° field angle?

3. Strehl ratio

a. Give the definition of the Strehl ratio.

b. Recall the Maréchal criterion, as defined on the root-mean square of the wavefront defect; at which value for the Strehl ratio does it correspond?

c. Recall the approximation of the root-mean square of the wavefront defect with the Zernike decomposition coefficients.

D. Measurements

You will study the shape and the dimension of the PSF by the point source method after the magnifying objective and use the HASO to directly measure the wavefront. You will also compare your direct observation of the PSF with the one simulated by the HASO software.

g. Direct analysis of the PSF on axis

► Switch on the laser diode (POWER, then ENBL ON) and adjust the current with the knob.

► Place the doublet lens in its best orientation and on axis; adjust its position with respect to the light source to ensure a transverse magnification gy = -1/3. Make sure that the beam fully illuminates the objective.

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► Look at the PSF on the camera with the microscope viewer; be careful and avoid saturation of the CMOS sensor. Choose a microscope objective that ensures a sufficient resolution in the image. Adjust the lens orientation to make the PSF rotationally symmetric (this adjustment will be improved with the HASO later).

► Measure the size of the PSF on axis by using the microscope viewer in the best focus plane.

Does it seem to you that this lens “diffraction limited” on axis at full aperture?

h. Study of the lens on axis at full aperture with the HASO

► Follow the HASO adjustment procedure above (see A.5). Make sure that the stray light is minimized.

► Observe the image spots on the HASO CCD and compare their dimensions to the PSF of individual micro-lenses (see C.1.a).

Evaluate roughly the number of micro-lenses that are illuminated and used for the wavefront measurement. This number is also given in the Slope window.

► Try to orient the lens on axis by observing and minimizing the Zernike coefficients associated to off-axis aberrations and minimizing the RMS amplitude of the wavefront defect σΔ.

► What is the numerical aperture measured by the HASO in these conditions? Compare to your theoretical estimation in C.2.b, and deduce an experimental evaluation of the actual magnifying ratio of the setup.

Call the professor to cross-check your adjustments.

► Observe the measured wavefront. What is the defect amplitude, PV and RMS? What are the main Zernike coefficients in the wavefront decomposition?

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Explain what FOCUS corresponds to and why FOCUS and TILT are not relevant to characterize the quality of the optical system under test.

IN THE FOLLOWING, ALWAYS SUBTRACT TILT AND FOCUS TO

MEASURE THE WAVEFRONT DEFECTS.

► Evaluate the optical path delay (PV and RMS) of the wavefront with respect to the reference sphere.

How does the RMS OPD compare to the Maréchal criterion?

What are the main aberrations of the lens on axis, i.e. the aberrations that have amplitudes larger than the noise of the wavefront decomposition? For each of the aberrations evaluate the Peak-to-Valley amplitude in the Seidel basis.

► Evaluate the experimental accuracy of your wavefront measurements, i.e. the amplitude of the wavefront defects and the Zernike coefficients, by moving the HASO around whilst remaining in acceptable operating conditions. The history knob ( ) of the RMS et PV values can be helpful to evaluate your experimental accuracy.

► Simulate the PSF of the objective on axis with the HASO software. Evaluate the size of the PSF by using the cross sections.

In which plane is the PSF calculated? Explain.

What is the Strehl ratio value? Comments.

? Compare this numerical simulation of the PSF to your direct measurement with the camera.

i. Fine study of the PSF through direct measurement

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The doublet adjustements being now using the HASO measurements, resume the direct analysis of the PSF as observed with the microscope viewer on the camera.

► Save an image of the PSF and evaluate the diameter encircling 84% of light energy with the MatLab script.

► Observe how the PSF changes when you defocus forth and back from the best focus plane. What can you conclude regarding the performance of this lens?

► Observe the evolution of the PSF and of its maximum irradiance while decreasing the lens diaphragm diameter.

j. Study of the magnifying objective off axis at full aperture

► Turn progressively the lens around the vertical axis and measure both the wavefront (HASO) and the PSF (microscope viewer) for 0.5°, 1° and 2°. Take special attention to the evolution of:

the different Zernike coefficients related to geometrical aberrations;

the PV and RMS values of the wavefront defect; the shape and dimension of the PSF.

Compare the simulations of the PSF deduced from the wavefront measurement to its direct measurement?

Is the evolution of the Zernike coefficients with the field angle consistent with the 3rd order theory?

► At a field angle of 2°, find for which diaphragm diameters – and thus which NA – the PSF is diffraction-limited, both based on the wavefront measurement and on the direct observation of the PSF.

Are these two measurements in agreement?

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► At a field angle of 2°, find at which diaphragm diameters the PSF diameter is the smaller. Does it correspond to a maximum irradiance of the PSF?

► Summarize your analyses obtained from both methods for the doublet lens under these operating conditions.

TP aberrations lab sessions 2016

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