introduction to waves; 1d and 3d wave equations; …...constant envelope e0 (amplitude) carrier...
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Paraxial focusing by a thin quadratic GRIN lens
d f
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rn(r)
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axial
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• Result:
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Spherical refractive surface withaxial index profile n(z)
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Gradient Index (GRIN) optics:
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Correction of spherical aberrationby axial GRIN lenses
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Generalized GRIN:what is the ray path through arbitrary n(r)?
material with variable
“optical path length”
lightray
optical “density”
P
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Let’s take a break from optics ...
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Mechanical oscillator
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Hamiltonian Optics postulates
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These are the “equations of motion,” i.e. they yield the ray trajectories.
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The ray Hamiltonian
The choice yields
Therefore, the equations of motion become
Since the ray trajectory satisfies a set of Hamiltonian equations on the quantity H, it follows that H is conserved.
The actual value of H=const. is
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=const. is arbitrary.
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The ray Hamiltonian and the Descartes sphere
p(s)n(q(s))
=0
The ray momentum pis constrained to lie on
a sphere of radius nat any ray location qalong the trajectory s
Application:Snell’s law of refraction
opticalaxis
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The ray Hamiltonian and the Descartes sphere
p(s)n(q(s))
=0
The ray momentum pis constrained to lie on
a sphere of radius nat any ray location qalong the trajectory s
Application:propagation in a GRIN medium
n(q)
The Descartes sphere radius is proportional to n(q); as the rays propagate, the lateral momentum is preserved by gradually changing the ray orientation to match the Descartes spheres.
Figure by MIT OpenCourseWare. Adapted from Fig. 1.5 in Wolf, Kurt B. Geometric Optics in Phase Space. New York, NY: Springer, 2004.
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physically allowable physically allowablekinetic energy refractive index
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Hamiltonian analogies:optics vs mechanics
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Example: Hamiltonian ray tracing of quadratic GRIN
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off-axis incidence, px(0)=0.2
Further reading: • M. Born and E. Wolf, Principles of Optics, Cambrige University Press, 7th edition, sections 4.1-4.2• K. B. Wolf, Geometrical Optics on Phase Space, Springer, chapters 1, 2 • K. Tian, Three-dimensional (3D) optical information processing, PhD dissertation, MIT 2006.
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So far
• Geometrical Optics – Light propagation is described in terms of rays and their
refraction / reflection – Image formation is described in terms of point sources and
point images • Region of validity of geometrical optics: features of interest should be
much bigger than the wavelength �– Problem: geometrical point objects/images are infinitesimally small,
definitely smaller than �!!!– So light focusing at a single point is an artifact of the geometric
approximations– Moreover, especially in microscopy, we are interested in resolving
object features at distances comparable to �– To understand light behavior at scales ~ � we need to graduate from
geometric to the wave description of light propagation, i.e. Wave Optics.
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What is a wave?
Image removed due to copyright restrictions. Please see:
http://www.reuters.com/news/pictures/searchpopup?picId=8505659
Water waves: Vouliagmeni Bay, Greece (photo by G. B.)
Photo of a Yagi-Uda antenna removed due to copyright restrictions.
Ground surface waves: Chocolate Hills, Philippines http://www.bohol.ph/picture48.html
Radio waves: Yagi-Uda television antenna Courtesy of Jeroen Hellingman. Used with permission.
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What is a wave?
• A wave is a traveling disturbance
• Evidence of wave behavior:
Interference Diffraction
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Traveling waves
Traveling disturbance Traveling disturbance (envelope) with sinusoidal modulation
(envelope on a carrier)
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Traveling wave terminology
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envelope
carrier
group velocity
phase velocity
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Harmonic wave
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constant envelope E0(amplitude)
carrier
phase velocity c
Dispersion relation
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The significance of phase delays
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Phase delays and interference
in phase out of phase
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