ECE 665 Fourier Optics

1
ECE 665Fourier Optics

• Spring, 2004
• Thomas B. Fowler, Sc.D.
• Senior Principal Engineer
• Mitretek Systems

2
Course goal

• To provide an understanding of optical systems
  for processing temporal signals as well as images
  
• Course is based on use of Fourier analysis in two
  dimensions to understand the behavior of optical
  systems

3
Nature of light and theories about it

• Fourier optics falls under wave optics
• Provides a description of propagation of light
  waves based on two principles
• Harmonic (Fourier) analysis
• Linearity of systems

Quantum optics
Electromagnetic optics
Wave optics
Ray optics
4
Course organization

• 13 weeks
• Main text Introduction to Fourier Optics, Joseph
  Goodman, McGraw-Hill, 1996
• Other material to be downloaded from Internet
• Student evaluation
• Homework 20
• Midterm exam 40
• Final exam 40

5
Topics

• Week 1 Review of one-dimensional Fourier
  analysis
• Week 2 Two-dimensional Fourier analysis
• Weeks 3-4 Scalar diffraction theory
• Weeks 5-6 Fresnel and Fraunhofer diffraction
• Week 7 Transfer functions and wave-optics
  analysis of coherent optical systems
• Weeks 8-9 Frequency analysis of optical imaging
  systems
• Week 10 Wavefront modulation
• Week 11 Analog optical information processing
• Weeks 12-13 Holography

6
Week 1 Review of One-Dimensional Fourier Analysis

• Descriptions time domain and frequency domain
• Principle of Fourier analysis
• Periodic series
• Sin, cosine, exponential forms
• Non-periodic Fourier integral
• Random
• Convolution
• Discrete Fourier transform and Fast Fourier
  Transform
• A deeper look Fourier transforms and functional
  analysis

7
Basic idea what you learned in undergraduate
courses

• A periodic function f(t) can be expressed as a
  sum of sines and cosines
• Sum may be finite or infinite, depending on f(t)
• Object is usually to determine
• Frequencies of sine, cosine functions
• Amplitudes of sine, cosine functions
• Error in approximating with finite number of
  functions
• Function f(t) must satisfy Dirichlet conditions
• Result is that periodic function in time domain,
  e.g., square wave, can be completely
  characterized by information in frequency domain,
  i.e., by frequencies and amplitudes of sine,
  cosine functions

8
Historical reason for use of Fourier series to
approximate functions

• Breaks periodic function f(t) into component
  frequencies
• Response of linear systems to most periodic waves
  can be analyzed by finding the response to each
  harmonic and superimposing the results)

9
Basic idea what you learned in undergraduate
courses (continued)

• Periodic means that f(t) f(tT) for all t
• T is the period
• Period related to frequency by T 1/f0 2?/?0
• ?0 is called the fundamental frequency
• So we have
• n?0 2n?/T is nth harmonic of fundamental
  frequency

10
How to calculate Fourier coefficients

• Calculation of Fourier coefficients hinges on
  orthogonality of sine, cosine functions
• Also,

11
How to calculate Fourier coefficients (continued)

• And we also need

12
How to calculate Fourier coefficients (continued)

• Step 1. integrate both sides
• Therefore

13
How to calculate Fourier coefficients (continued)

• Step 2. For each n, multiply original equation by
  cos nw0t and integrate from 0 to T

0
0
0
Therefore
14
How to calculate Fourier coefficients (continued)

• Step 3. Calculate bn terms similarly, by
  multiplying original equation by sin nw0t and
  integrating from 0 to T
• Get similar result
• Some rules simplify calculations
• For even functions f(t) f(-t), such as cos t,
  bn terms 0
• For odd functions f(t) -f(-t), such as sin t,
  an terms 0

15
Calculation of Fourier coefficients examples

• Square wave (in class)

1
T
T/2
-1
16
Calculation of Fourier coefficients examples
(continued)
Gibbs phenomenon ringing near discontinuity

• Result

Source http//mathworld.wolfram.com/FourierSeries
.html
17
Calculation of Fourier coefficients examples
(continued)

• Triangular wave (in class)

V
T
T/2
-V
18
Calculation of Fourier coefficients examples
(continued)

• Triangle wave result
• Note that value of terms falls off as inverse
  square

19
Other simplifying assumptions half-wave symmetry

• Function has half-wave symmetry if second half is
  negative of first half

20
Other simplifying assumptions half-wave symmetry

• Can be shown

21
Conditions for convergence

• Conditions for convergence of Fourier series to
  original function f(t) discovered (and named for)
  Dirichelet
• Finite number of discontinuities
• Finite number of extrema
• Be absolutely convergent
• Example of periodic function excluded

22
Parseval's theorem

• If some function f(t) is represented by its
  Fourier expansion on an interval -l,l, then
• Useful in calculating power associated with
  waveform

23
Effect of truncating infinite series

• Truncation error function en(t) given by
• This is difference between original function and
  truncated series sn(t), truncated after n terms
• Error criterion usually taken as mean square
  error of this function over one period
• Least squares property of Fourier series states
  that no other series with same number n of terms
  will have smaller value of En

24
Effect of truncating infinite series (continued)

• Problem is that there is no effective way to
  determine value of n to satisfy any desired E
• Only practical approach is to keep adding terms
  until En lt E
• One helpful bit of information concerns fall-off
  rate of terms
• Let k number of derivatives of f(t) required to
  produce a discontinuity
• Then
• where M depends on f(t) but not n

25
Some DERIVE scripts

• To generate square wave of amplitude A, period T
• squarewave(A,T,x) Asign(sin(2pix/T))
• For Fourier series of function f with n terms,
  limits c, d
• Fourier(f,x,c,d,n)
• Example Fourier(squarewave(2,2,x),x,0,2,5)
  generates first 5 terms (actually 3 because 2 are
  zero)
• To generate triangle wave of amplitude A, period
  T
• int(squarewave(A,T,x),x)
• Then Fourier transform can be done of this

26
Exponential form of Fourier Series

• Previous form
• Recall that

27
Exponential form of Fourier Series (continued)

• Substituting yields
• Collecting like exponential terms and using fact
  that 1/j -j

28
Exponential form of Fourier Series (continued)

• Introducing new coefficients
• We can rewrite Fourier series as
• Or more compactly by changing the index

29
Exponential form of Fourier Series (continued)

• The coefficients can easily be evaluated

30
Exponential form of Fourier Series (continued)

• Sometimes coefficients written in real and
  complex terms as
• where

31
Exponential form of Fourier Series example

• Take sawtooth function, f(t) (A/T)t per period
• Then
• Hint if using Derive, define w 2p/T, set
  domain of n as integer

32
Fourier analysis for nonperiodic functions

• Basic idea extend previous method by letting T
  become infinite
• Example recurring pulse

v0
t
a/2
-a/2
T
33
Fourier analysis for nonperiodic functions
(continued)

• Start with previous formula
• This can be readily evaluated as

34
Fourier analysis for nonperiodic functions
(continued)

• Using fact that T 2p/w0, may be written
• We are interested in what happens as period T
  gets larger, with pulse width a fixed
• For graphs, a 1, V0 1

35
Effect of increasing period T
a/T
a/T
a/T
36
Transition to Fourier integral

• We can define f(jnw0) in the following manner
• Since difference in frequency of terms Dw w0 in
  the expansion. Hence

37
Transition to Fourier integral (continued)

• Since
• It follows that
• As we pass to the limit, Dw -gt dw, nDw -gt w so we
  have

38
Transition to Fourier integral (continued)

• This is subject to convergence condition
• Now observe that since
• We have

39
Transition to Fourier integral (continued)

• In the limit as T -gt ?
• Since f(t) 0 for t lt -a/2 and t gt a/2
• Thus we have the Fourier transform pair for
  nonperiodic functions

40
Example pulse

• For pulse of area 1, height a, width 1/a, we have
• Note that this will have zeros at w 2anp,
  n0,1, 2
• Considering only positive frequencies, and that
  most of the energy is in the first lobe, out to
  2ap, we see that product of bandwidth 2ap and
  pulse width 1/a 2p

41
Example of pulse
5
width1
1
-1/2
1/2
1/10
-1/10
width0.2
42
Pulse limiting cases

• Let a -gt ?, then f(t) -gt spike of infinite height
  and width 1/a (delta function) -gt 0
• Transform -gt line F(jw)1
• Thus transform of delta function contains all
  frequencies
• Let a -gt 0, then f(t) -gt infinitely long pulse
• Transform -gt spike of height 1, width 0
• Now let height remain at 1, width be 1/a
• Then transform is

43
Pulse limiting cases (continued)

• Now, we are interested in limit as a -gt 0 for w
  -gt 0 and w gt 0
• First, consider case of small w
• So when a -gt 0, 1/a -gt ?
• As w moves slightly away from 0, it drops to zero
  quickly because of w/2a term in denominator
  (numerator lt1 at all times)
• So we get delta function, d(0)

44
Fourier transform of pulse width 0.1
45
Properties of delta function

• Definition
• Area for any g
  gt 0
• Sifting property
• since

46
Some common Fourier transform pairs
Source http//mathworld.wolfram.com/FourierTransf
orm.html
47
Some Fourier transform pairs (graphical
illustration)
transform
function
transform
function
Source Physical Optics Notebook Tutorials in
Fourier Optics, Reynolds, et. al., SPIE/AIP
48
Fourier transform Gaussian pulses
49
Properties of Fourier transforms

• Simplification
• Negative t
• Scaling
• Time
• Magnitude

50
Properties of Fourier transforms (continued)

• Shifting
• Time convolution
• Frequency convolution

51
Convolution and transforms

• A principal application of any transform theory
  comes from its application to linear systems
• If system is linear, then its response to a sum
  of inputs is equal to the sum of its responses to
  the individual inputs
• This was original justification for Fourier's
  work
• Because a delta function contains all frequencies
  in its spectrum, if you hit something with a
  delta function, and measure its response, you
  know how it will respond to any individual
  frequency
• The response of something (e.g., a circuit) to a
  delta function is called its impulse response
• Called point spread function in optics
• Often denoted h(t)

52
Convolution and transforms (continued)

• The Fourier transform of the impulse response can
  be calculated, usually designated H(jw)
• Therefore if one knows the frequency content of
  an incoming signal u(t), one can calculate the
  response of the system
• The response to each individual frequency
  component of incoming signal can be calculated
  individually as product of impulse response and
  that component
• Total response is obtained by summing all of
  individual responses
• That is, response Y(jw) H(jw)U(jw)
• Where U(jw) is sum of Fourier transforms of
  individual components of u(t)

53
Convolution and transforms (continued)

• May be visualized as

U(jw)
H(jw)
Y(jw)H(jw)U(jw)
Input
System
Response
54
Convolution and transforms (continued)

• Example
• Signal is square wave, u(t)sgn(sin(x))
• This has Fourier transform
• So response Y(jw) is

55
Convolution and transforms (continued)

• If incoming signal described by Fourier integral
  instead, same result holds
• To get time (or space) domain answer, we need to
  take inverse Fourier transform of Y(jw)

56
Convolution and transforms (continued)

• Can also be calculated in time (or space), i.e.,
  non-transformed domain
• Derivation
• Now, we introduce new variables v and t, related
  to t and z by

57
Convolution and transforms (continued)

• Computing Jacobean to transform variables
• Implies that differential areas same for both
  systems of variables
• Thus since t v-z v-t we have
• Where we have calculated the limits as follows

58
Convolution and transforms (continued)

• We may assume without loss of generality that
  u(z) 0 for zlt0
• Otherwise we can shift variables to make it so
• Must assume that u(z) has some starting point
• Therefore the lower limit of integration in the
  inner integral is 0
• We may also assume without loss of generality
  that h(t) 0 for tlt0
• Therefore h(v-t) 0 for t gt v

59
Convolution and transforms (continued)

• Since the outer integral defines a Fourier
  transform, its inverse is just y(t), so we have
• This is usually written with t as the inner
  variable,
• This is called the convolution of h and u,
  usually written y(t) hu
• Can readily be calculated on a computer

60
Convolution old way (graphically)

61
Convolution old way (continued)
Source P. S. Rha, SFSU, http//online.sfsu.edu/
psrha/ ENGR449_PDFs/EE449_L5_Conv.PDF
62
Convolution and transforms (new way)

• Use computer algebra programs
• Some Derive scripts
• Step function u(t)if(tlt0,0,1)
• Pulse of width d, amplitude a f1(t)if(tgt0 and
  tltd,a,0)
• Triangle of width d, amplitude a
  triangle(t)if(tgt0 and tltd/2,2at/d,(if(tgtd/2
  and tltd,2a-2at/d,0)0)
• Convolution convolution(t)int(f1(t-t)f2(t),t,0
  ,t)
• Example
• f1 is pulse of width 1, amplitude 1
• f2 is pulse of width 2, amplitude 3

63
Convolution functions
64
Convolution useful web sites

• http//www.jhu.edu/signals/
• http//mathworld.wolfram.com/Convolution.html
• http//www.annauniv.edu/shan/Lap1.1.9.html
• http//rivit.cs.byu.edu/morse/550-F95/node12.html

65
Fourier and Laplace transforms

• Fourier transform does not preserve initial
  condition information
• Therefore most useful when steady state
  conditions exist
• This is typically the case for optical systems
• But often not true for electrical networks
• Comparison of definitions

Laplace
Fourier
66
Fourier and Laplace transforms (continued)

• Differences
• In Fourier transform, jw replaces s
• Limits of integration are different, one-sided
  vs. two-sided
• Contours of integration in inverse transform
  different
• Fourier along imaginary axis
• Laplace along imaginary axis displaced by s1
• Conversion between Fourier and Laplace transforms
• Laplace transform of f(t) Fourier transform of
  f(t)e-st
• Symbolically,

67
Fourier transforms of random sources (noise)

• Noise has frequency characteristics
• Generally continuous distribution of frequencies
• Since transform of individual frequencies gives
  spikes, this allows us to separate signal from
  noise via Fourier methods
• Common types of noise
• White noise equal power per Hz (power doubles
  per octave)
• Pink noise equal power per octave
• Other colors of noise described at
  http//www.hoohahrecords.com/resfreq/articles/nois
  e.html
• Fourier transform distinguishes these

68
Fourier transforms of random sources (noise)
(continued)

• Frequency domain thus allows us to obtain
  information about signal purity that is difficult
  to obtain in time (or space) domain
• Noise
• Distortion

69
Fourier transforms of random sources (noise)
(continued)

Source http//hesperia.gsfc.nasa.gov/schmahl/fou
rier_tutorial/node6.html
70
Discrete and Fast Fourier Transforms

• Most Fourier work today carried out by computer
  (numerical) analysis
• Discrete Fourier transform (DFT) is first step in
  numerical analysis
• Simply sample target function f(t) at appropriate
  times
• Replace integral by summation
• Here tn nT, where Tsampling interval, N
  number of samples, and frequency sampling
  interval W 2p/NT, wk kW

71
Discrete and Fast Fourier Transforms (continued)

• Sampling frequency fs 1/T
• Frequency resolution Df 1/NT fs/N
• For accurate results, sampling theorem tells us
  that sample frequency fs gt 2 x fmax, the highest
  frequency in the signal
• Implies that highest frequency captured fmax lt
  1/2T fs/2
• Otherwise aliasing will occur
• To improve resolution, note that you can't double
  sampling frequency, as that also doubles N (for
  same piece of waveform)
• The only way to increase N without affecting fs
  is to increase acquisition time

72
Discrete and Fast Fourier Transforms (continued)

• Note that DFT calculation requires N separate
  summations, one for each wk
• Since each summation requires N terms, number of
  calculations goes up as N 2
• Therefore doubling frequency resolution requires
  quadrupling number of calculations
• Method also assumes function f(t) is periodic
  outside time range (nT) considered
• Also note that raw DFT calculation gives array of
  complex numbers which must be processed to give
  usual magnitude and phase information
• When only power information required, squaring
  eliminates complex terms

73
Inverse discrete Fourier transform

• Calculated in straightforward manner as
• This gives, of course, the original sampled
  values of the function back
• Other values can be determined by appropriate
  filtering

74
Uses of DFT

• DFT usage may be visualized as

Power Spectral Density
Power Spectrum
Magnitude
Phase
DFT Spectrum
75
Power measurements and DFT

• Power spectrum
• Gives energy (power) content of signal at a
  particular frequency
• No phase information
• Squared magnitude of DFT spectrum

76
Power spectral density

• Derived from power spectrum
• Generally normalized in some fashion to show
  relative power in different ranges
• Measures energy content in specific band

77
Fast Fourier Transform (FFT)

• Developed by Cooley and Tukey in 1965 to speed up
  DFT calculations
• Increases speed from O(N2) to O(N log N), but
  there are requirements
• Useful reference http//www.ni.com/swf/presentati
  on/us/fft/

78
Fast Fourier Transform (FFT) (continued)

• Requirements for FFT
• Sampled data must contain integer number of
  cycles of base (lowest frequency) waveform
• Otherwise discontinuities will exist, giving rise
  to spectral leakage, which shows up as noise
• Signal must be band limited and sampling must be
  at high enough rate
• Otherwise aliasing occurs, in which higher
  frequencies than those capturable by sampling
  rate appear as lower frequencies in FFT
• Signal must have stable (non-changing) frequency
  content
• Number of sample points must be power of 2

79
Spectral leakage
No discontinuities
Discontinuities present

Source National Instruments
80
Fast Fourier Transform (FFT) (continued)

• We will not discuss exactly how the method works
• Lots of software packages are available
• See this site for many of them http//ourworld.com
  puserve.com/homepages/steve_kifowit/fft.htm
• Contained in Mathcad package
• Also available in many textbooks
• Many modern instruments such as digital
  oscilloscopes have FFT built-in
• Averaging is frequently used to improve result
• Averages over several FFT runs with different
  data sets representing same waveform
• Sometimes with slightly staggered start times

81
FFT (continued)

• Also inverse FFT exists for going in opposite
  direction
• Short Mathcad demo
• Note that output of FFT is two-dimensional array
  of length ½ number of sample points 1
• The points in this array are the complex values
  F(jwk)
• But the wk values themselves do not appear
• Must be calculated by user
• They are wk k x frequency resolution k x
  2p/NT, k 0...N/2

82
FFT examples showing different resolution
f(x)sin (px/5), analysis done in MATHCAD
64 sample points, T1 sec, fs1resolution 1/64 Hz
32 sample points, T1 sec, fs1resolution 1/32 Hz
83
Fourier analysis a deeper view

• Fourier series only one possible way to analyze
  functions
• Best understood in terms of functional analysis
• Let X be a space composed of real-valued
  functions on some interval a,b
• Technically, the set of Lebesgue-integral
  functions
• Infinite-dimensional space
• Define an inner product (dot product in
  Euclidean space) as follows

84
Fourier analysis a deeper view (continued)

• This induces a norm on the space
• Can be shown that this space is complete
• Complete normed space with norm defined by inner
  product is known as a Hilbert space
• An orthogonal sequence (uk) is a sequence of
  elements uk of X such that

85
Fourier analysis a deeper view (continued)

• This series can be converted into an orthonormal
  sequence (ek) by dividing each element uk by its
  norm uk
• Consider an arbitrary element x ? X, and
  calculate
• Now formulate the sum
• Then clearly if x-xn?0 as n??, the sum
  converges to x

86
Fourier analysis a deeper view (continued)

• We have the following theorem If (ek) is an
  orthonormal sequence in Hilbert space X, then
• (a) The series converges (in the
  norm on X) if and only if the following series
  converges
• (b) If the series converges, then the
  coefficients ak are the Fourier coefficients
  so that x can be written

87
Fourier analysis a deeper view (continued)

• (c) For any x ? X, the foregoing series converges
• Lemma Any x in X can have at most countably many
  (may be countably infinite) nonzero Fourier
  coefficients with respect to an
  orthonormal set (ek)
• Note that we are not quite where we want to be
  yet, as we have not shown that every x ? X has a
  sequence which converges to it
• For this we require another notion, that of
  totality

88
Fourier analysis a deeper view (continued)

• Note also that as of this point we have said
  nothing about the nature of the functions ek
• Any set which meets the orthogonality condition
  is
  OK, since it can be normalized
• Note that (sin nt), (cos nt) meet condition, can
  be combined into new set containing all elements
  by suitable renumbering
• Lots of other functions would work as well, such
  as triangle waves, Bessel functions

89
Fourier analysis a deeper view (continued)

• Most interesting orthonormal sets are those which
  consists of sufficiently many elements so that
  every element in the space can be approximated by
  Fourier coefficients
• Trivial in finite-dimensional spaces just use
  orthonormal basis
• More complicated in infinite dimensional spaces
• Define a total orthonormal set in X as a subset M
  ? X whose span is dense in X
• Functions analogously to orthonormal basis in
  finite spaces
• But Fourier expansion doesn't have to equal every
  element, just get arbitrarily close to it in
  sense of norm

90
Fourier analysis a deeper view (continued)

• Can be shown that all total orthonormal sets in a
  given Hilbert space have same cardinality
• Called Hilbert dimension or orthogonal dimension
  of the space
• Trivial in finite dimensional spaces
• Necessary and sufficient condition for totality
  of an orthonormal set M is that there does not
  exist a non-zero x ? X such that x is orthogonal
  to every element of M

91
Fourier analysis a deeper view (continued)

• Parseval relation can be expressed as
• Another theorem states that an orthonormal set M
  is total in X if and only if the Parseval
  relation holds for all x
• True for (sin nt)/p, (cos nt)/p terms
• Therefore these terms form total orthonormal set
• Key results
• Fourier expansion works because (sin nt)/p, (cos
  nt)/pterms from orthonormal basis for space of
  functions
• Any other orthonormal set of functions can also
  serve as basis of Fourier analysis

92
Fourier analysis a deeper view (continued)

• Effect of truncating Fourier expansion
• Finite set (e1...em) no longer total
• But it can be shown that the projection theorem
  applies

Function f(x) to be approximated
Approximation error
Approximation fm(x)
Space spanned by (e1...em)
93
Fourier analysis a deeper view (continued)

• Projection theorem states that optimal
  representation of f(x) in lower-order space
  obtained when error f fm is orthogonal to
  fm
• This is guaranteed by orthonormal elements ei and
  the construction of the Fourier coefficients
• Therefore truncated Fourier representation is
  optimal representation in terms of (e1...em)
• References
• Erwin Kreyszig, Introductory Functional Analysis
  with Applications
• Eberhard Zeidler, Nonlinear Functional Analysis
  and its Applications, Vol. I, Fixed-Point Theorems