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. 3rd 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
(respectively α") become large, Abbe’s formula (aplanatic system) nysinα =n"y"sinα" is not valid anymore.In other words, coma corresponds to the fact that the magnification ratio gy = 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 3rd 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#
=
Δcoma by 3cos
Figure 1: 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 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 3rd order astigmatism is given by:
ϕ
The wavefront departure associated to the field curvature is given by:
2
27 The tangential rays, i.e. the rays that remain in the incidence plane1 (azimuth angle
=0
ϕ ), 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 2). 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 3). 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:
2 C y s S !2
= and
2 C y t T !2
=
Figure 2: 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.
1 The incidence plane is defined by the object and the optical axis of the lens.
28
Figure 3: 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 CT, CS, and C; the latter is called the field curvature. Here, CT, CS, and C are negative quantities, while the astigmatism curvature A=(CS −CT)/2 is positive.
The focal line lengthes, sagittal and tangential, observed in S and T are :
2!!′!!′!"# with A!=! C! −C! 2 and the diameter of the spot is !!′!!′!"#. For a thin optical system, one can show that
'
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:
λ = 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 θ, which is the field angle. (see Figure 4).
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.
29 Figure 4: 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 5: 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.
C. Preliminary calculations
θ
y’
f’ = 500 mm
Optical system
collimator
Hole source LED
Optical axis
Input pupil Microscope viewer
30
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 angleθ = 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?
D. 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.
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
31
► 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 θ = 0° and measure the defocus distances of the sagittal2, tangential3 and least scatter planes with respect to the paraxial plane, for
0 ° ≤ θ ≤ 10 °
by a step of 2°, and also for θ = -5°. 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.