Aberrations Labworkinphotonics-Semester8

Labwork in photonics - Semester 8

Aberrations

General presentation

Introduction . . . 1

The point source method . . . 3

Wavefront measurements . . . 5

Specifications of the lenses . . . 11

The point source method Lab 1 On-axis aberrations. . . 15

Lab 2 Off-axis aberrations . . . 25

Wavefront mesurements Lab 3 Zygo wavefront analysis by interferometric measurements . . . . 33 Lab 4 Haso Shack-Hartmann wavefront analysis . . . . 51

Engineer 2^nd year - Palaiseau Version: February 5, 2018 Year 2017-2018

Baptiste FIX

David HOLLEVILLE

Lionel JACUBOWIEZ

ii

Guidelines

All the information about Travaux Pratiques (TP) is posted on the bulletin 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 lab partner has a major problem preventing their attendance, the other should 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 attend will both have to deposit a written justification at the secretary’s office, and also warn directly the teacher in charge of the stu- dents’lab of the reason for his/her non attendance (before the session, if the absence is predictable).

Justified absence. Coming back to do the missed lab The stu- dent must always contact the lab session instructors to envisage the possibility of coming back to do the missed lab. This possi- bility will depend on the availability of the instructor, the equip- ment and the lab rooms. If agreed, the student then comes back to do the 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 rerun lab on behalf of the availability of the students’ lab service, then this lab will not be redone nor graded (the average for the labs will be done using all the other sessions). This lab will still remain part of the programme for the final lab exam

iii

iv

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

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

If the student refuses the proposed date for the rerun lab, he will be 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 revision sessions. In case of repeted absences, the teacher in charge of that school year (1st, 2nd or 3rd ) will not allow the student to attend the first lab exam session organized at the end of the school year.

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

Punctuality

You are expected to arrive on time for the lab sessions. A student who is very or frequently late will not be allowed to complete the lab and the 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. Examples include:

— Copying word for word (or nearly so) from a book or Web page without putting the passage in quotes and mentioning its source.

— Inserting in your work graphs or images coming from external sources (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).

v

— Summarizing an original idea of another author using his or her words 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 to the "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 therefore strictly forbidden to bring food or drink into the entire lab area, includ- ing 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 regular lab hours, see the TP staff , Thierry Avignon or Cédric Lejeune (office S1.18).

vi .

1

GENERAL PRESENTATION OF THE ABERRATIONS LAB SESSIONS

Updated January 2017

I NTRODUCTION

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 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.

2

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

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.

3

T HE 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:

4

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.

5

W AVEFRONT 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 P_i(u,ϕ) used by the ZYGO software). Note that the polynomials P_i(u,ϕ) do not have the same norm. they can be normalized by multiplying them by the factors k_i indicated in Table 1, i.e. the polynomials Z_i(u,ϕ)=k_i ×P_i(u,ϕ) constitute an orthonormal basis over the disc pupil, with Z_i(u,ϕ)² =π.

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

) , u ( P C _i

i

i × ϕ

=

Δ

∑

The RMS OPD is thus evaluated as follows:

6

∑

Δ = σ

0 i

i2 2i

k C

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 3^rd 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 Δ_max = 6×C₈¹ Δ_PV = Δ_max

Coma _{Δmax = 3 C6}² _+C7² Δ_PV = 2× Δ_max

Astigmatism _{Δmax = C4}² _{+ C5}² _{ΔPV = 2× Δmax}

Polynomials used in ZyMOD /PC and HASO

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

Ci k_i P_i(u,ϕ)=R_n(u)cosmϕ

C0 1 1 Piston

C1 2 ucosϕ

Tilt

C2 2 usinϕ

C3 3 2u² −1 ^Defocus

C4 6 u²cos2ϕ

Astigmatism 3 rd order

C5 6 u²sin2ϕ

C6 8 (3u² −2)ucosϕ

Coma C7 8 (3u² −2)usinϕ

C8 5 6u⁴−6u²+1 _Spherical

1Note 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 3^rd order: Δmax = 6 C8 – 30 C15. This can be extrapolated easily to the other terms.

7 aberration

C9 8 u³cos3ϕ

Trefoil

5 th order

C10 8 u³sin3ϕ

C11 10 (4u²−3)u²cos2ϕ

Astigmatism C12 10 (4u² −3)u²sin2ϕ

C13 12

(

)

Coma C14 12 (10u⁴ −12u² +3)usinϕ

C15 7 20u⁶−30u⁴+12u²−1 ^Spherical aberration

C16 10 u⁴cos4ϕ

Tetrafoil

7 th order C17 10 u⁴sin4ϕ

C18 12 (5u²−4)u³cos3ϕ

Trefoil C19 12 (5u²−4)u³sin3ϕ

C20 14 (15u⁴−20u²+6)u²cos2ϕ

Astigmatism C21 14 (15u⁴−20u² +6)u²sin2ϕ

C22 16 (35u⁶ −60u⁴+30u²−4)ucosϕ

Coma C23 16 (35u⁶−60u⁴ +30u²−4)usinϕ

C24 9 70u⁸−140u⁶+90u⁴−20u²+1 ^Spherical aberration C25 12 u⁵cos5ϕ

Pentafoil

9 th order C26 12 u⁵sin5ϕ

C27 14 (6u² −5)u⁴cos4ϕ

Tetrafoil C28 14 (6u² −5)u⁴sin4ϕ

C29 16 (21u⁴−30u²+10)u³cos3ϕ ^Trefoil

8

C30 16 (21u⁴−30u²+10)u³sin3ϕ

C31 18 (56u⁶ −105u⁴ +60u²−10)u²cos2ϕ

Astigmatism C32 18 (56u⁶ −105u⁴ +60u²−10)u²sin2ϕ

C33 20 (126u⁸−280u⁶+210u⁴−60u² +5)ucosϕ

Coma C34 20 (126u⁸−280u⁶+210u⁴−60u² +5)usinϕ

C35 11 252u¹⁰ −630u⁸+560u⁶−210u⁴+30u²−1 ^Spherical aberration

C36 13 924u¹² −2772u¹⁰ +3150u⁸−1680u⁶+420u⁴−42u²+1 ^Spherical aberration

11 thorder

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,

(

)

Δ . Assuming that the pupil is uniformly illuminated, the amplitude of the electromagnetic field in the pupil is related to the wavefront defects by:

λ πΔ ϕ −

=

=

) y , x 2 ( j 0 ) y , x ( j 0 P P

p p p

p a e

e a ) y , x ( P

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

( ) ( ( ) ( ) ) ( ( ) )

!!"

#

$$%

&

λ' '

!!"

#

$$%

&

λ '

' =

⊗ ' =

x r x

r

2 P P P

P P

P,y P x ,y FTP x ,y

x P FT r

PSF! _! ^!

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:

9

€

1,22× λ sinα $ _.

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 R_S ≥0.8. 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 λ = 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.

( ) ( )

(

)

(

( )

)

MTF P

0

r = − !

⊗

= ⊗

ω!! ^{λ !⋅ω!} !

!

! !

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 ν, measured in the image plane (see Figure 3).

10

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

) ) ( 1 )

cos(

Arc 2( ) (

FTI ²

c c

c

diff ν

− ν ν

− ν ν

ν

= π ν

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:

(

)

=

ν_c 2sin _max /

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

For ν≥ν_c^{, MTF}

( )

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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S PECIFICATIONS 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 sinα"_max =0.08

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; sinα"_max =0.08

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

12

A ^PPENDIX 1 : S IMULATED CHARACTERISTICS OF THE PLANO -

CONVEX SINGLET , IN THE BEST ORIENTATION

Glass: BK7, Effective focal length: 150 mm, Numerical aperture: 0.08 (Ø_e = 25 mm) Spot diagram on axis at λ = 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 λ = 633 nm

Paraxial focus plane Minimum scatter focus plane

1 2

Longitudinal chromatism 3

4 5

13

A PPENDIX 2: S IMULATED CHARACTERISTICS OF THE

T HORLABS DOUBLET AC254-150-A1

Effective focal length: 150 mm, Numerical aperture: 0.08 (Øe = 25 mm) 8

Surface data 9

Doublet AC254-150-A1 10

SRF RADIUS THICKNESS APERTURE RADIUS GLASS SPE NOTE 11

OBJ -- 1.0000e+20 8.7489e+18 AIR 12

AST 91.620000 5.700000 12.500000 A N-BK7 C 13 14

2 -66.680000 2.200000 12.500000 P SF5 C 15

3 -197.700000 146.135443 S 12.500000 P AIR 16

IMS -- -- 13.121179 S 17 18

Ray intercept curves (ask the professor for details) 19 20

21 22 23

Decomposition of the wavefront on the Zernike polynomials basis, θ = 5°, λ = 633 nm 24

*ZERNIKE ANALYSIS Best focus 25

WAVELENGTH 1 26

Positive angle A is a rotation from the +y axis toward the +x axis. 27

-0.051376: [0] 1 28

-0.042076: [1] RCOSA 29

-- : [2] RSINA 30

-0.020738: [3] 2R^2 - 1 31

2.834575: [4] R^2COS2A 32

-- : [5] R^2SIN2A 33

-0.057919: [6] (3R^2 - 2)RCOSA 34

-- : [7] (3R^2 - 2)RSINA 35

0.026119: [8] 6R^4 - 6R^2 + 1 36

-0.010290: [9] R^3COS3A 37

-- : [10] R^3SIN3A 38

0.000781: [11] (4R^2 - 3)R^2COS2A 39

-- : [12] (4R^2 - 3)R^2SIN2A 40

0.011123: [13] (10R^4 - 12R^2 + 3)RCOSA 41

-- : [14] (10R^4 - 12R^2 + 3)RSINA 42

-0.004587: [15] 20R^6 - 30R^4 + 12R^2 – 1 43

44

λ = 656nm λ = 486nm

λ = 587nm

λ = 587nm

14

45

15

LAB SESSION N°1: ON-AXIS ABERRATIONS THE “POINT SOURCE METHOD”

Updated January 2017

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 labeledin 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 λ. 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 Δn of the index over the visible domain. In fact, instead of Δn, we use the constringency (also called Abbe number), defined as , which is a useful quantity to calculate the chromatic aberrations:

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

The available glasses may be represented in a graph (ν, n) as shown on Figure 1.

Remember that the constringency characterizes a dielectric material, not an optical system.

ν=

(

)

ν

= n

− 1

n

− n

16

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: δf’ = f’_B – f’_R = - f’/νd.

Figure 1: different glasses in the (ν, 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 3^rd 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: where α^’ is the image aperture angle

- the radius of the PSF at the paraxial focus is given by:

- 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 :

l ( ) α ! ^{= F}

^{! F}

^!

^{= a α !}

dy!=−l

( )

Flints

17 Best orientation (curved side towards infinity):

(

)

n 2

2 n 2

a n^{3 2} ₂ !

− +

= −

Worst orientation (curved side towards the focus):

(

)

2

a n² 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 λ=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

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.

18

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.

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

Grating

Flat mirrors Iodine lamp

Input hole Output hole

19 length f!=H!F! of the lens under test evolves when you vary the wavelength

(to

study chromatism). The measured distance z!=H!A! is related to f! through:

f 1 D

1 z

1

= !

! +

Variations in z! and f! are thus related through:

D z f 1 2 D z

1 f f

2

Δ !

#$

& % '

( !

−

!≅

# Δ

$

& % '

( !

−

!= Δ

The measured variations in z! should be corrected by the factor _!

"

$ #

%

& '

− D

f

1 2 in order to

evaluate the corresponding variations in the focal length.

C.

Preparatory Questions

Simple lens aberrations

1. Calculate the Airy spot diameter of the lens (λ = 546 nm).

2. Evaluate the primary axial chromatism ^Δf!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

ΔA! to deduce the evolution of the focal point ΔF!(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 ?

20

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.

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 λ=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 3^rd 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. Compare your observations to the 3^rd 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.

21

► 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 (λ = 486nm) and the C line (λ = 656nm) of hydrogen (used in the definition of constringency (Abbe number)).

► Plot F_C"F_λ" versus λ. 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 F_C"F_λ" .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.

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

22

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 λ = 635nm. The single mode fiber (type SM600) has a mode diameter of 4.3µm and a numerical aperture of ~0.10 at λ = 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.

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.

23

► 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

24

25

LAB SESSION N°2: OFF- AXIS ABERRATIONS THE “POINT SOURCE METHOD”

Updated January 2017

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).

A. Memo

1. 3^rd order coma

Coma is an off-axis aberration that leads to a PSF that resembles a comet (see Figure 1)

Coma is related to the fact that, when the field dimension y (respectively y!) and the aperture angle

α

In other words, coma corresponds to the fact that the magnification ratio g_y = ^y!_y

varies with the aperture angle α".

This happens only for off-axis objects, with height y (or angular height θ 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 α", also depends on the angular position of the ray intercept in the pupil plane, i.e. on the azimuth angle ϕ.

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

α"

"

=

ρ by sin²

where ‘b’ denotes the coma parameter. The length of the coma PSF is 3ρ (see

26

Figure 1). The associated wavefront defect is given by:

ϕ

#α#

=

Δ_coma by ³cos

Figure 1: Shape of the PSF associated with 3^rd 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 θ 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 3^rd order astigmatism is given by:

ϕ α

=

Δ y' ' cos2 4

A _{2 2}

ast , where

( )

2 C A C^S − ^T

= .

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

2

curv y'2 '

4

C α

−

=

Δ , where

( )

2 C C C^S + ^T

= .