Compound Optics and Prisms Tuesday, 9/26/2006 Physics 158 Peter Beyersdorf
Document info
10. 1
Class Outline Lens and mirror systems Aberrations Stops Prisms
10. 2
Aberrations Chromatic Monochromatic Spherical Coma Astigmatism Field Curvature Distortion 10. 3
Chromatic Aberration Due to dispersion in glass lenses, different wavelengths will be bent by different amounts and be focus at different locations. This can be compensated with a doublet using a suitable pair of materials for the two lenses 10. 4
Spherical Aberrations Deviation of a lens from the ideal shape of a cartesian oval (particularly great at large radial distances) gives rise to different focal points for rays at different radial positions Transverse spread of a focused spot is called transverse spherical aberration (TSA) Longitudinal spread of a focused spot is called Longitudinal spherical aberration (LSA) The circle of least confusion is the location of minimal spot size 10. 5
Coma Abberation from image “plane” not actually being plane Off axis light creates a comet-like blurring of the image
10. 6
a
'l
EF * *
E S
^ a
s 4
i i
,EA
Distortions of off-axis rays due to the difference in incident geometry of the sagittal and meridional rays
g3
Astigmatism
10. 7
Field Curvature Aberration from object “plane” not being planar
10. 8
Distortion Aberration from the positional dependance on the transverse magnification of a lens
undistorted
barrel distortion pincushion distortion 10. 9
Lens and Mirror Systems Consider a system of multiple imaging elements The image from one element can be the object for another Multiple elements may be necessary to minimize certain aberrations (i.e. by compensating for the aberrations of one optic with those of another) Various apertures in the system will limit the field of view and angular acceptance of the system 10.10
Stops and Pupils The element that ultimately limits how much light enters an optical system is the aperture stop for that system The element that limits the field of < 3
n
the image is called the field stop The image of the aperture stop seen from the object (at the optical axis) is called the entrance pupil
LJ
Fr
The image of the aperture stop seen from the image location (at the optical axis) is called the exit pupil
10. 11
Aperture Stops and Pupils Find the aperture stop, entrance and exit pupil for the following compound system as seen from the object 80mm before the first lens
x=-80 mm
x=0 mm Φ=50 mm f=100 mm
x=75 mm Φ=25 mm
x=100 mm Φ=25 mm f=50 mm
10.12
Aperture Stops and Pupils Find the aperture stop, entrance and exit pupil for the following compound system as seen from the object 80mm before the first lens
Ent. Pupil
Exit Pupil AS
+
x=-80 mm
x=0 mm Φ=50 mm f=100 mm
+ x=75 mm Φ=25 mm
x=100 mm Φ=25 mm f=50 mm
x=∞
10.13
Numerical Aperture The amount of light a lens collects is a function of its numerical aperture, approximately the ratio of its diameter to its focal length
N A = n0 sin θaccept
D ≈ f
n0
θaccept
The inverse quantity is called the f/# of a lens
D
f
f f /# = D
A high numerical aperture is said to be a “fast lens”. Why? A camera lens has a series of F/#s that form a geometric series (f/1, f/1.4, f/2, f/2.8, f/4 …) why?
14
10.
Numerical Aperture What are the benefits of having a fast lens (high numerical aperture) Lots of light collection → bright image Tightly focused spot (i.e. less diffraction)
What are the benefits of having a slow lens (low numerical aperture) Reduced spherical aberation Good depth of field 15
10.
Prisms Prisms have a wide range of different applications in optics. index measurement devices retro-reflectors optical retarders wavelength based separators periscopes beam deflectors beam splitters
10.16
Minimum Deviation Angle From Snell’s law δ
and from symmetry
so that
giving
from inspection 10.17
Minimum Deviation Angle The deviation angle δ is δ
and so
giving
and from Snell’s law
This is a common way to measure very accurately the index of refraction of a material
10.18
Prism Examples Brewster prism
Littrow prism
mirror surface
Brewster’s angle at the entrance is n=tan θi1. For glass n≈1.5 so θi1≈56° This type of prism is used in laser cavities as wavelength selection devices.
Two Littrow prisms back-toback can be used as a transmissive wavelength selective device. For glass of n=1.5, α=68° 10.19
Prism Examples Direct vision prism This is a color separator that produces an undeviated ray for a certain wavelength
Constant deviation prism The Pellin-Broca prism This prism has a geometry such that an outgoing ray exits at 90° relative to the incoming ray. 45° 30° 45° 90° 60°
Typically the center piece is flint glass and the two outer components are crown glass
30°
90°
60°
λ0 It is a single block of glass but can be viewed a three separate triangles consisting of two 30°-60° triangles joined by a 45°-45° triangle.
10.20
More Prism Examples The right angle prism
The Amici prism
rh rh
lh
rh
Total internal reflection reflects 100% of the light and introduces a reversion to the image
It works like a right-angle prism, but the roof-top shape at the hypotenuse adds a reversion to the image
10.21
More Prism Examples The Porro prism
Dove prism lh rh
rh
rh
If we rotate the prism, the image rotates at twice the rate
Two TIR reflections. Image reversion from right-angle prism is cancelled. Hence output has the same handedness as the input 10.22
More Prism Examples The Penta prism
Causes a 90°. deviation without affecting the orientation of the image
Corner cube
Uses three bounce to reflect light back into the direction it came from
10.23
More Prism Examples Rhomboid prism
rh
rh
A special case of the rhomboid prism is the Fresnel rhomb, which is used to generate a phase shift of the orthogonal polarizations to produce circularly polarized light for 45° linearly polarized input light. Two successive rhombs produce a half-waveplate effect .
10.24
More Prism Examples Double Porro prism
Prism beam expander
w
rh
W rh
Used in binoculars for image erection.
For a single prism the magnification is
When θti=α we get maximum magnification 10.25
Example Problem Show that in a prism beam expander, the magnification along one axis of the image is
w
W
while along the other axis the magnification is unity
10.26
Example Problem Show that in a prism beam expander, the magnification along one axis of the image is
w
W
while along the other axis the magnification is unity
10.27
Summary Real world imaging systems suffer from various aberrations Finite lens sizes lead to limits on the amount and angle of light collected Compound lenses can be used to minimize distortions of improve the amount of angle of light collected. Prisms perform a variety of imaging functions
