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=
(
n −1)
Δnν
d= n
d− 1
n
F− n
C16
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 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: 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
P! F
α!
!= a α !
2dy!=−l
( )
α! × !α =−aα!3 CrownsFlints
17 Best orientation (curved side towards infinity):
(
n 1)
fWorst orientation (curved side towards the focus):
(
n 1)
f2
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.
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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
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The measured variations in z! should be corrected by the factor !
"
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 ?
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