Transcript PPT - Jung Y. Huang

An Overview on the Huygens-Fresnel
Principle, Coherence and van CitterZernike Theorem
References:
1.Modeling and Simulation of Beam Control Systems: Part 1. Foundations of Wave
Optics Simulation www.mza.com/publications/MZADEPSBCSMSCP3.ppt
2.Professor David Attwood (Univ. California at Berkeley), AST 210/EECS 213,
Lecture 16, http://ast.coe.berkeley.edu//sxreuv/2005/Ch08C.pdf
1. Huygens-Fresnel Principle of Wave
Propagation
Fundamental theory of wave propagation
•
•
•
Wave equation (monochromatic) in
vacuum or uniform dielectric medium
(1)
Wave equation in presence of
fluctuations n(x,y,z; t): third term
couples the polarizations during
propagation
Fundamental approximation: order of
magnitude calculations imply that the
coupling term is negligible.
In this approx., the fluctuations do not mix
polarization components
 Turbulent prop still satisfies the
“scalar diffraction” picture.
Resulting equation, with extra
decomposition n(r) = <n>+dn(r), and
letting k = k0 n0 = averaged wave vector
in unperturbed medium
(2)
(3)
perturbation term relative
to Eq (1)
Scalar Diffraction Theory
When monochromatic light propagates through vacuum or ideal dielectric
media, the spatial and temporal variations of the electromagnetic field can
be separated, and the spatial variations of the six components of the
electric and magnetic field vectors are identical. The spatial variation
of the two vector fields, E and B, can therefore be represented in terms of
a single scalar field, u.
electromagnetic field
scalar field
Non-monochromatic light can be
expressed as a superposition of
monochromatic components:
The Huygens-Fresnel Principle
The propagation of optical fields is described by the Huygens-Fresnel
principle, which can be stated as follows:
Knowing the optical field over any given plane in vacuum or an ideal
dielectric medium, the field at any other plane can be expressed as a
superposition of “secondary” spherical waves, known as Huygens
wavelets, originating from each point in the first plane.
u2
u1
R
r1
r2
q
z1
z2
Huygens
wavelets
The Fresnel Approximation
When the transverse extents of the optical field to be propagated
are small compared with the propagation distance, we can make
small angle approximations, yielding useful simplifications.
u2
u1
r1
R
r2
q
z1
z2
The Fresnel Approximation
Conditions for Validity
The Fresnel approximation is based upon the assumption |r2-r1| << Dz.
Here r1 and r2 represent the transverse coordinates in the initial and final
planes for any pair of points to be considered in the calculation. What
pairs of points must be considered depends upon the specific problem to
be modeled.
This requirement will be satisfied if the transverse extents of the region
of interests at the two planes are sufficiently small, as compared to the
propagation distance.
The requirement can also be satisfied if the light is sufficiently wellcollimated, regardless of the propagation distance.
The Fresnel approximation can also be used, in a modified form, for light
that is known to approximate a known spherical wave, such as the light
propagating between the primary and secondary mirrors of a telescope.
Fourier Optics
When the Fresnel approximation holds, the Fresnel diffraction integral can
be decomposed into a sequence of three successive operations:
1. Multiplication by a quadratic phase factor
2. A scaled Fourier transform
3. Multiplication by a quadratic phase factor.
quadratic
phase
factor
scaled
Fourier
transform
quadratic
phase
factor
The Fourier Transform
Physical Interpretation of the Fresnel
Diffraction Integral
u2
u1
r2=0
r1=0
z1
z2
Equivalently,
the quadratic
phaseappearing
factors can
The
two quadratic
phase factors
in be
the
thought diffraction
of as two Huygens
wavelets, to
originating
Fresnel
integral correspond
two
from the points
(r1=0, z=z1) and (r2=0, z=z2).
confocal
surfaces.
Fourier Optics in Operator Notation
For notational convenience it is sometimes useful to express Fourier
optics relationships in terms of linear operators. We will use PDz to
indicate propagation, FDz for a scaled Fourier transform, and QDz
for multiplication by a quadratic phase factor.
Multi-Step Fourier Propagation
z1
It is sometimes useful to carry
out a Fourier propagation in
two or more steps.
Dz
z2
The individual propagation
steps may be of any size and
in either direction.
z1
z2
Fourier Optics: Some Examples
(all propagations between confocal planes)
circ
rect(x)rect(y)
Gaussian
FDz
FDz
FDz
Airy pattern
sinc(x)sinc(y)
Gaussian
Waves vs. Rays
Scalar diffraction theory and Fourier optics are usually described in
terms of waves, but they can also be described, with equal rigor, in
terms of rays.
This may seem surprising, because rays are constructs more
typically associated with geometric optics, as opposed to wave optics.
In geometric optics, rays are thought of as carrying an energy,
possibly distributed over a range of wavelengths. In wave optics,
each ray must be thought of as carrying a certain complex
amplitude, at a specific wavelength.
The advantage of thinking in terms of rays, as opposed to waves, is
that it makes it easier to take into account geometric considerations,
such as limiting apertures. A wave can be thought of as a set of
rays, and geometric considerations may allow us to restrict our
attention to a smaller subset of that set.
A Wave as a Set of Rays
u2
u1
r2
r1
z1
z2
Suppose
we
collect
all
impinging
onlight
the
point
zcan
from
all
Repeating
Each
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Each
Huygen’s
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procedure
wavelet
the
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can
forthe
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be
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points
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adecomposed
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wave
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wave
ra1 set
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decomposed
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ar
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waves
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(Huygen’s
z withwavelets)
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alla points
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on
at
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the
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1zon
2. Conversely,1 the same ray also
originating
originating
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plane
all
alloriginating
zthe
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aon
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2. points
2 to the
Waves vs. Rays
Mathematical Equivalence
Ray picture:
that the
Note that the field at each point r2 isRecall
expressed
as “wave
the picture”
equations
were
superposition of the contributions from
all points
r1. derived from
the “ray picture” equation with
no additional assumptions.
Wave picture:
Note that the field u2 at all points is expressed
in terms of the field u1 at all points.
Waves vs. Rays
Why the “Ray Picture” is Useful
Thinking of light as being made up of rays, as opposed
to waves, makes it easier to take into account a priori
geometric constraints pertaining to two or more planes at
the same time.
For example, if the light to be modeled is known to pass
through a limiting apertures, we can restrict our attention
to just the set of the rays that pass through that aperture.
Similarly, if there are multiple limiting apertures, we can
restrict our attention to the intersection of the ray sets
defined by the individual apertures.
It is important to understand that strictly speaking a
given ray set remains well-defined only within a
contiguous volume filled with a uniform dielectric
medium, and only for purely monochromatic light.
Extending Scalar Diffraction Theory
Relatively easy / cheap
Monochromatic
Coherent
Uniform polarization
Ideal media
a
a
a
a
Quasi-monochromatic
Temporal partial coherence
Non-uniform polarization
Phase screens, gain screens
Harder / more expensive
•
•
•
•
Broadband illumination
Spatial partial coherence
Ultrashort pulses
Wide field incoherent imaging
Scalar Diffraction Theory and Fourier
Optics
Scalar Diffraction Theory: the electric and magnetic vector
fields are replaced by a single complex-valued scalar field u.
The Huygens-Fresnel Principle: knowing the field at any plane,
the field at any other plane can be expressed as a superposition
of spherical waves originating from each point in the first plane.
The Fresnel Approximation: for |r|<<|Dz|, the equations simplify.
Fourier Optics: the propagation integral can be expressed in
terms of Fourier transforms and quadratic phase factors.
Waves vs. Rays: light waves can be thought of as sets of rays,
where each ray carries a complex amplitude.
Extending Fourier Optics: it is possible.
The Discrete Fourier Transform
What happens when we try to represent a continuous
complex field on a finite discrete mesh?
How can we reconstruct the continuous field from the discrete
mesh?
How can we ensure that the results obtained will be correct?
What can go wrong?
Reference: The Fast Fourier Transform, by Oran Brigham
The DFT as a Special Case of the Fourier Transform
F
window
F
sample
F
repeat
F
DFT pair
The DFT as a Special Case of the Fourier Transform
u
u  rect
repeat (u ×rect)
F
F
F
Constructing the Continuous Analog of a DFT Pair
uD
One
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Now way
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have
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doorder
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to
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a function
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To
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each
iteration,
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spacing as
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and
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1
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New DFT Pair
Constructing the Continuous Analog of a DFT Pair
Example: A Discrete “Point Source”
N=16
u
F(u)
N=64
N=256
The Nyquist Criterion
The Whitaker-Shannon Sampling Theorem shows that it is possible to
exactly recover a continuous function from a discretely sampled version of
that function if and only if (a) the function is strictly band-limited and (b)
the sample spacing satisfies the Nyquist Criterion: the spacing must be
less than or equal to half the period of the highest frequency component
present.
In the context of wave optics simulation the Nyquist criterion defines the
maximum mesh spacing that will suffice to represent a given optical
field:
Here qmax is the band-limit of the complex field to be represented on the
discrete mesh when we compute the DFT in the course of performing a
DFT propagation. Note that this step occurs only after we have multiplied
the field by a quadratic phase factor:
The Nyquist Criterion
Wave Optics Example
u1
θmax
θmax
D1
2θmax
1
u2
2
1
2θmax
D2
2
d1
d2
z1
z2
Aliasing
If we attempt to represent a field with energy propagating
at angles exceeding the Nyquist limit for the given mesh
spacing, that energy will instead show up at angles below
the Nyquist limit; this phenomenon is called aliasing.
The Discrete Fourier Transform
What happens when we try to represent a continuous complex field
on a finite discrete mesh?
We lose any energy falling outside the mesh extents in either
domain. Discrete sampling in one domain implies periodicity in
the other.
How can we reconstruct the continuous field from the discrete mesh?
DFT interpolation. (Or, to obtain a new DFT pair, a somewhat more
complicate procedure involving two DFT interpolations.)
How can we ensure that the results obtained will be correct?
By enforcing the Nyquist criterion.
What can go wrong?
Aliasing
2. Optical Coherence and van CitterZernike Theorem
principle
q  tan 1 (r z )