Title: ECE 665 Fourier Optics
1ECE 665Fourier Optics
- Spring, 2004
- Thomas B. Fowler, Sc.D.
- Senior Principal Engineer
- Mitretek Systems
2Course 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
3Nature 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
4Course 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
5Topics
- 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
6Week 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
7Basic 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
8Historical 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)
9Basic 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
10How to calculate Fourier coefficients
- Calculation of Fourier coefficients hinges on
orthogonality of sine, cosine functions - Also,
11How to calculate Fourier coefficients (continued)
12How to calculate Fourier coefficients (continued)
- Step 1. integrate both sides
- Therefore
13How 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
14How 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
15Calculation of Fourier coefficients examples
1
T
T/2
-1
16Calculation of Fourier coefficients examples
(continued)
Gibbs phenomenon ringing near discontinuity
Source http//mathworld.wolfram.com/FourierSeries
.html
17Calculation of Fourier coefficients examples
(continued)
- Triangular wave (in class)
V
T
T/2
-V
18Calculation of Fourier coefficients examples
(continued)
- Triangle wave result
- Note that value of terms falls off as inverse
square
19Other simplifying assumptions half-wave symmetry
- Function has half-wave symmetry if second half is
negative of first half
20Other simplifying assumptions half-wave symmetry
21Conditions 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
22Parseval'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
23Effect 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
24Effect 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
25Some 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
26Exponential form of Fourier Series
- Previous form
- Recall that
27Exponential form of Fourier Series (continued)
- Substituting yields
- Collecting like exponential terms and using fact
that 1/j -j
28Exponential form of Fourier Series (continued)
- Introducing new coefficients
- We can rewrite Fourier series as
- Or more compactly by changing the index
29Exponential form of Fourier Series (continued)
- The coefficients can easily be evaluated
30Exponential form of Fourier Series (continued)
- Sometimes coefficients written in real and
complex terms as - where
31Exponential 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
32Fourier analysis for nonperiodic functions
- Basic idea extend previous method by letting T
become infinite - Example recurring pulse
v0
t
a/2
-a/2
T
33Fourier analysis for nonperiodic functions
(continued)
- Start with previous formula
- This can be readily evaluated as
34Fourier 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
35Effect of increasing period T
a/T
a/T
a/T
36Transition to Fourier integral
- We can define f(jnw0) in the following manner
- Since difference in frequency of terms Dw w0 in
the expansion. Hence
37Transition to Fourier integral (continued)
- Since
- It follows that
- As we pass to the limit, Dw -gt dw, nDw -gt w so we
have
38Transition to Fourier integral (continued)
- This is subject to convergence condition
- Now observe that since
- We have
39Transition 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
40Example 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
41Example of pulse
5
width1
1
-1/2
1/2
1/10
-1/10
width0.2
42Pulse 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
43Pulse 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)
44Fourier transform of pulse width 0.1
45Properties of delta function
- Definition
- Area for any g
gt 0 - Sifting property
- since
46Some common Fourier transform pairs
Source http//mathworld.wolfram.com/FourierTransf
orm.html
47Some Fourier transform pairs (graphical
illustration)
transform
function
transform
function
Source Physical Optics Notebook Tutorials in
Fourier Optics, Reynolds, et. al., SPIE/AIP
48Fourier transform Gaussian pulses
49Properties of Fourier transforms
- Simplification
- Negative t
- Scaling
- Time
- Magnitude
50Properties of Fourier transforms (continued)
- Shifting
- Time convolution
- Frequency convolution
51Convolution 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)
52Convolution 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)
53Convolution and transforms (continued)
U(jw)
H(jw)
Y(jw)H(jw)U(jw)
Input
System
Response
54Convolution and transforms (continued)
- Example
- Signal is square wave, u(t)sgn(sin(x))
- This has Fourier transform
- So response Y(jw) is
55Convolution 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)
56Convolution 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
57Convolution 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
58Convolution 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
59Convolution 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
60Convolution old way (graphically)
61Convolution old way (continued)
Source P. S. Rha, SFSU, http//online.sfsu.edu/
psrha/ ENGR449_PDFs/EE449_L5_Conv.PDF
62Convolution 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
63Convolution functions
64Convolution 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
65Fourier 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
66Fourier 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,
67Fourier 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
68Fourier 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
69Fourier transforms of random sources (noise)
(continued)
Source http//hesperia.gsfc.nasa.gov/schmahl/fou
rier_tutorial/node6.html
70Discrete 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
71Discrete 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
72Discrete 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
73Inverse 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
74Uses of DFT
- DFT usage may be visualized as
Power Spectral Density
Power Spectrum
Magnitude
Phase
DFT Spectrum
75Power measurements and DFT
- Power spectrum
- Gives energy (power) content of signal at a
particular frequency - No phase information
- Squared magnitude of DFT spectrum
76Power spectral density
- Derived from power spectrum
- Generally normalized in some fashion to show
relative power in different ranges - Measures energy content in specific band
77Fast 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/
78Fast 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
79Spectral leakage
No discontinuities
Discontinuities present
Source National Instruments
80Fast 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
81FFT (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
82FFT 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
83Fourier 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
84Fourier 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
85Fourier 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
86Fourier 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
87Fourier 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
88Fourier 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
89Fourier 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
90Fourier 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
91Fourier 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
92Fourier 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)
93Fourier 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