GG 711: Advanced Techniques in Geophysics and Materials Science

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GG 711  Advanced Techniques in Geophysics and
Materials Science
Lecture 6 Fourier Optics and 3-D Image
Formation in Confocal Microscopy
Pavel Zinin HIGP, University of Hawaii,
Honolulu, USA
www.soest.hawaii.edu\zinin
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Lecture Overview

• Introduction to Fourier Transform
• Fourier Spectrum Approach of Image in 3-D
• Contrast in Reflection and Transmission
  Microscopy
• Emulated Transmission Confocal Raman Microscopy

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The Fourier Transform 
What is the Fourier Transform?   The continuous
limit the Fourier transform (and its inverse) 
Jean Baptiste Joseph Fourier
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The Fourier Transform
Consider the Fourier coefficients. Lets define
a function F(?) that incorporates both cosine and
sine series coefficients, with the sine series
distinguished by making it the imaginary
component where t is the time and ? is the
angular frequency ? 2 ? f, where f is the
frequency
The Fourier Transform
F(?) is called the Fourier Transform of f(t). It
contains equivalent information to that in f(t).
We say that f(t) lives in the time domain, and
F(?) lives in the frequency domain. F(?) is
just another way of looking at a function or wave.
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The Inverse Fourier Transform
The Fourier Transform takes us from f(t) to F(?).
How about going back?
Inverse Fourier Transform
Fourier Transform Notation There are several
ways to denote the Fourier transform of a
function.  If the function is labeled by a
lower-case letter, such as f, we can write  
Lord Kelvin Fouriers theorem is not only one
of the most beautiful results of modern analysis,
but it may be said to furnish an indispensable
instrument in the treatment of nearly every
recondite question in modern physics.
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What do we hope to achieve with the Fourier
Transform?
We desire a measure of the frequencies present in
a wave. This will lead to a definition of the
term, the spectrum.
Plane waves have only one frequency, w.
This light wave has many frequencies. And the
frequency increases in time (from red to blue).
It will be nice if our measure also tells us when
each frequency occurs.
From Prof. Rick Trebino, Georgia Tech
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The Fourier Transform and its Inverse
FourierTransform  
Inverse Fourier Transform
So we can transform to the frequency domain and
back. Interestingly, these functions are very
similar.   There are different definitions of
these transforms. The 2p can occur in several
places, but the idea is generally the same.
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Fourier Transform of a rectangle function
rect(t)
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Sinc(x) and why it's important

• Sinc(x) is the Fourier transform of a rectangle
  function.
• Sinc2(x) is the Fourier transform of a triangle
  function.
• Sinc(x) describes the axial field distribution of
  a lens.
• Sinc2(ax) is the diffraction pattern from a slit.
• It just crops up everywhere...

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Fourier Transform of a rectangle function
rect(t)
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A sum of sines and cosines
Gallanger et al., AJR. 190. 2008
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Time-Resolved Acoustic Microscope
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Sampling and Nyquist Theorem
The NyquistShannon sampling theorem is a
fundamental result in the field of information
theory, in particular telecommunications and
signal processing. Sampling is the process of
converting a signal (for example, a function of
continuous time or space) into a numeric sequence
(a function of discrete time or space). The
theorem states
Theorem If a function x(t) contains no
frequencies higher than B hertz, it is completely
determined by giving its ordinates at a series of
points spaced 1/(2B) seconds apart.
In order for a band-limited (i.e., one with a
zero power spectrum for frequencies f gt B)
baseband ( f gt 0) signal to be reconstructed
fully, it must be sampled at a rate f ? 2B . A
signal sampled at f 2B is said to be Nyquist
sampled, and f 2B is called the Nyquist
frequency. No information is lost if a signal is
sampled at the Nyquist frequency, and no
additional information is gained by sampling
faster than this rate.
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Fourier Optics
Fourier optics is the study of classical optics
using techniques involving Fourier transforms and
can be seen as an extension of the
Huygens-Fresnel principle. The plane wave
spectrum concept is the basic foundation of
Fourier Optics. The plane wave spectrum is a
continuous spectrum of uniform plane waves, and
there is one plane wave component in the spectrum
for every tangent point on the far-field phase
front. The amplitude of that plane wave component
would be the amplitude of the optical field at
that tangent point.
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Light Waves Plane Wave
Definition A plane wave is a wave in which the
wavefront is a plane surface a wave whose
equiphase surfaces form a family of parallel
surfaces (MCGraw-Hill Dict. Of Phys., 1985).
Definition A plane wave in two or three
dimensions is like a sine wave in one dimension
except that crests and troughs aren't points, but
form lines (2-D) or planes (3-D) perpendicular to
the direction of wave propagation (Wikipedia,
2009).
The large arrow is a vector called the wave
vector, which defines (a) the direction of wave
propagation by its orientation perpendicular to
the wave fronts, and (b) the wavenumber by its
length. We can think of a wave front as a line
along the crest of the wave.
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The Fourier Transform in Space 
Time Fourier Transform
Fourier Transform in Space?
As ? must satisfy the Helmholtz equation, every
solution can be decomposed into plane waves
exp(ikxx ikyy ikzz) with wave vector k (kx,
ky, kz), k?/c, and c being the velocity of
sound in the coupling liquid. If kx2 ky2 ? k²,
then . Such waves are denoted as homogeneous
waves.
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The Fourier Transform in Space 
Gallanger et al., AJR. 190. 2008
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The Fourier Transform in Space Angular Spectrum  
Conversely, the potential can then be written as
the inverse Fourier transform of the angular
spectrum.
The inverse Fourier transform represents the
wavefield in the plane Z as a superposition of
plane waves exp(ikxx ikyy).
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Properties of the Fourier Transform
The Fourier spectrum of the field at the plane z
Z, U(kx,ky,Z) can be expressed through the
Fourier spectrum of the field at the plane z 0,
U(kx,ky,0).
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Properties of the Fourier Transform
The Fourier spectrum of the circular Function
.
Is the jinc function
J. W. Goodman. Introduction to Fourier Optics,
, McGraw-Hill, (1996).
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Image Formation in 3-D
Reciprocity Principle and Signal of the Optical
and Acoustical Microscopes
Atalar applied Auld's reciprocity principle to
derive the output voltage V of the detetor of the
reflection SAM in terms of angular spectra on an
arbitrary plane z
where Ui(kx, ky) is the angular spectrum of the
field the microscope detector emits into the
coupling liquid, Us(kx, ky) is the angular
spectrum of the field resulting from scattering
of the field incident on the object. It is the
spectrum of the field the microscope detects. In
Equation a proportionality constant is omitted.
The position of the plane z at which the spectra
are evaluated is the same for both but z can be
chosen arbitrarily.
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Field in the Focus and Fourier Spectrum Approach
In the limit f ? 8 (Debye approximation) this
equation is valid throughout the whole space. It
expresses the field in the focal region as a
superposition of plane waves whose propagation
vectors fall inside a geometrical cone formed by
drawing straight lines from the edge of the
aperture through the focal point, which, in
contrast to the usual Debye integral, are
weighted with P(?,?). Where P(?,?) is the Pupil
Function.
We may say that each point on P(?,?) is
responsible for the emission (and, by
reciprocity, for the detection) of the plane wave
component emitted along the line from the point
on P(?,?) through the focal point
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Field in the Focus and Fourier Spectrum Approach
To find the angular spectrum of the emitted field
in the focal plane, we set the third coordinate
of r equal to zero and substitute Cartesian
coordinates for the angular integration
variables. For the vector components of k hold
kx k sin? cosf, ky k sin? sinf, kz cosf
. The Jacobi determinant
corresponding yields.
Denoting the Cartesian coordinates of r by x, y,
z, Equation can be rewritten
.
(1)
Up to a constant P(kx,ky) is defined as
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Field in the Focus and Fourier Spectrum Approach
.
Comparison with of the Fourier integral shows
that this is the angular spectrum of the
transducer field in the focal plane provided that
?i(x,y,0) has the same Fourier transform as the
transducer field in the focal plane.
Equation (1) also tells us that if a transmissive
object is placed one focal length in front of a
lens, then its Fourier transform will be formed
one focal length behind the lens.
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Field in the Focus and Fourier Spectrum Approach
The Pupil Function of an ideal lens can be
described by a circle function
Fourier transform of the circle function
.
J. W. Goodman. Introduction to Fourier Optics,
, McGraw-Hill, (1996).
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Field in the Focus and Fourier Spectrum Approach
Equation (1) also tells us that the lens performs
Fourier transform on the incident electromagnetic.
Plane wave passing the lens with a pupil function
P(x,y) transforms into jinc function in focal
plane of the lens.
.
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Lens as a Fourier transform
.
Courtesy to Vitali. Prakapenka, University of
Chicago
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Lens as a Fourier transform
Fourier transform of letter E
.
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Field in the Focus and Fourier Spectrum Approach
Now we know the spectrum incident on the object
and we want to calculate the scattered spectrum.
Consider imaging of a three-dimensional object as
depicted in Figure 6. In the case of a reflection
microscope, the same transducer serves as emitter
and as detector, while for the transmission
microscope the detector is situated confocally
across from the emitter. We will use three
coordinate systems, (x,y,z) with origin in the
focal point and (x',y',z') in point O' linked to
the particle. The z- and z'-axis are directed
away from the emitter (Figure 7). The coordinates
of the system O' are (X,Y,Z) from the focal
point. We introduce also an intermediate
coordinate system with its origin in point O''
with vector (x,y,z').
The spectrum Ui of the incident field as well as
the spectrum Us of the reflected field are also
given in the spatial coordinate system that
originates at O'. Since the magnitude of k is
fixed, the wave vector is completely specified by
the vector kt (kx, ky) kx, ky denote the wave
vector components of the scattered field, and
kx, ky the wave vector components of the
incident field. To calculate the spectrum of the
scattered field we must integrate over all
incident spectral components.
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Field in the Focus and Fourier Spectrum Approach
For the reflection microscope we have , since the
backscattered wave propagates opposite to the
z-direction. Combining all above we obtain the
output signal of the reflection microscope as
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Field in the Focus and Fourier Spectrum Approach
The spectrum Ui of the incident field as well as
the spectrum Us of the reflected field are also
given in the spatial coordinate system that
originates at O'. To calculate the spectrum of
the scattered field we must integrate over all
incident spectral components.
gs is the scattering function. Now we have to
determine the reflected spectrum with the help of
the incident spectrum in system O, and then to
express also the reflected spectrum in system O.
Let us first express the spectrum Ui incident in
the focal plane system O in system O. The
spectrum Ui incident on the object plane at
position Z is calculated by including a
propagation factor exp(ikzZ)
For the reflection microscope we have , since the
backscattered wave propagates opposite to the
z-direction. Combining all above we obtain the
output signal of the reflection SAM as
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Signal from a half space
Knowing the general expression for the output
signal of the SAM, we can derive the expression
of the images of simply shaped objects. We start
with the simplest case, the SAM signal from a
homogeneous half space. In this case, we can
present the spectrum of the reflected field as a
simple multiplication of the incident field and
the reflection coefficient.56
Combining two above equations we obtain for
scattering function
R(kz,ky) is the reflecton coefficient. Then
In spherical coordinate system
The contrast of the image in the reflection
microscope is determined by the reflection
coefficient of the object.
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Image Formation of Three-Dimensional Objects
In general the part of the scattering function
describing the behaviour of homogeneous
(non-evanescent) waves, which is important for
confocal imaging, can also be obtained directly
from the far-field distribution of the scattered
wave. The asymptotic behaviour of the field
distribution ?s scattered from a bounded obstacle
for an incident plane wave with wave vector k'
can be written as
, as kr ? 8.
where r is the distance from the origin and the
far-field scattering amplitude. The
relation of the homogeneous wave part of the
scattering function gs to the far-field
scattering amplitude
for kx2 ky2 lt k²
can be obtained using the angular spectrum
representation of the far-field distribution
Combining above equations and omitting the
normalizing constant we obtain
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Image Formation of Spherical Particles
The far-field angular distribution for scattering
by a spherical particle can be found analytically
where the An describe the scattering amplitudes
which depend on particle size and material
properties, and Pn are ordinary Legendre
polynomials. ?k is the angle between k and k'. In
the scalar theory the Anm are independent of m
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Image Formation of Spherical Particle in
Reflection Microscope
X-Z scan through a steel sphere for a reflection
microscope(OXSAM) at 105 MHz. The radius of the
sphere was 560 µm. The semi-aperture angle of the
microscope lens was 26.5. Z0 corresponds to
focussing to the centre of the sphere. (b) X-Z
scan through a steel sphere for reflection
microscope calculated with the same parameter as
in From Zinin, P. et al., Optik 107, 45, 1997.
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Some Fourier transform pairs (graphical
illustration)
X-z scan of 3-D image of steel sphere.
X-z scan of 3-D image of a liquid (water) drop.
X-z scan of 3-D image of a Plexiglas sphere.
Important conclusion from the theory we developed
is that size of the spherical particle can be
determined only from image taken by a
transmission microscope. The size of the image of
the spherical particle in reflection microscope
is less than the real size of the particle and is
equal to a sin(a), where a is the radius of the
particle, and a is semiaperure angle of the lens.
This theory has found several direct application
in practical microscopy surface imaging and in
developing Emulated Transmission Confocal Raman
Microscopy.
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Optical images of yeast cells
Optical images of yeast cells on glass (100x
objective) in transmission mode (a), in
reflection mode (b). The red circles mark the
position of the laser beam.
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Optical images of yeast cells
Calculated vertical scans through a transparent
sphere with refraction index of 1.05 and
refraction index of surrounding liquid of 1.33
(a) reflection microscope with aperture angle 30
o (a) transmission microscope with aperture
angle 30o.
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Optical images of yeast cells
Sketch of the optical rays when cell is (a)
attached to the glass substrate or (b) to mirror.
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Emulated Transmission Confocal Raman Microscopy
Optical image of the yeast bakery cells in the
reflection confocal microscope. Rectangle shows
the area of the Raman mapping. (a) Raman spectra
of the cell a measured with green laser
excitation (532 nm, WiTec system).
Map of the Raman peak intensity centered at 2933
cm-1. The intensity of the 2933 cm-1 peak is
shown in a yellow color scale. (b) Map of the
Raman peak intensity centered at 1590 cm-1. The
intensity of the 1590 cm-1 peak is shown in a
green color scale.
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Summary Lecture 6

• Sinc function is a Fourier Transform of the
  rectangula Function.
• Distribution of the field in the focal plane is
  the spacial Fourier Transform of the Pupil
  Function of the Lens
• Fourier Spectrum Approach of Image in 3-D
• Contrast in Reflection and Transmission
  Microscopy
• Emulated Transmission Confocal Raman Microscopy

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Reading Lecture 6

• J. W. Goodman. Introduction to Fourier Optics,
  Second Edition,, McGraw-Hill, (1996).
• Reynolds, et. al., The New Physical Optics
  Notebook Tutorials in Fourier Optics, SPIE,
  1989
• G. Fowles, Introduction to Modern Optics, 2nd
  edition, Dover, 1989
• T. Wilson and C. J. R. Sheppard . Theory and
  Practice of Scanning Acoustic Microscopy.
  Academic Press. (1984)
• P. Zinin and W. Weise, Theory and applications
  of acoustic microscopy, in T. Kundu ed.,
  Ultrasonic Nondestructive Evaluation Engineering
  and Biological Material Characterization. CRC
  Press, Boca Raton, chapter 11, 654-724 (2004).
• P. Zinin, W. Weise, O. Lobkis and S. Boseck, The
  theory of three dimensional imaging of strong
  scatterers in scanning acoustic microscopy. Wave
  Motion, 25(3), 212-235 (1997).
• W. Weise, P. Zinin, T. Wilson, G. A. D. Briggs
  and S. Boseck, Imaging of spheres with the
  confocal scanning optical microscope. Optics
  Letters. 21(22), 1800-1802 (1996).