1. 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

2. 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

3. 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

4. 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

5. 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)

6. Basic idea: what you learned in undergraduate courses (continued)
Periodic means that f(t) = f(t+T) 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

7. How to calculate Fourier coefficients
Calculation of Fourier coefficients hinges on orthogonality of sine, cosine functions
Also,

8. How to calculate Fourier coefficients (continued)
And we also need

9. How to calculate Fourier coefficients (continued)
Step 1. integrate both sides:
Therefore

10. How to calculate Fourier coefficients (continued)
Step 2. For each n, multiply original equation by cos nw0t and integrate from 0 to T:
Therefore

11. 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

12. Calculation of Fourier coefficients: examples
Square wave (in class) 1
T/2 T -1

13. Calculation of Fourier coefficients: examples (continued)
Gibbs phenomenon:
ringing near discontinuity
Result
Source:

14. Calculation of Fourier coefficients: examples (continued)
Triangular wave (in class) +V
T/2 T -V

15. Calculation of Fourier coefficients: examples (continued)
Triangle wave result
Note that value of terms falls off as inverse square

16. Other simplifying assumptions: half-wave symmetry
Function has half-wave symmetry if second half is negative of first half:

17. Other simplifying assumptions: half-wave symmetry
Can be shown

18. 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

19. 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

20. 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

21. 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 < 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

22. Some DERIVE scripts To generate square wave of amplitude A, period T:
squarewave(A,T,x) := A*sign(sin(2*pi*x/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

23. Exponential form of Fourier Series
Previous form
Recall that

24. Exponential form of Fourier Series (continued)
Substituting yields
Collecting like exponential terms and using fact that 1/j = -j:

25. Exponential form of Fourier Series (continued)
Introducing new coefficients
We can rewrite Fourier series as
Or more compactly by changing the index

26. Exponential form of Fourier Series (continued)
The coefficients can easily be evaluated

27. Exponential form of Fourier Series (continued)
Sometimes coefficients written in real and complex terms as
where

28. 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

29. 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

30. Fourier analysis for nonperiodic functions (continued)
Start with previous formula:
This can be readily evaluated as

31. 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

32. Effect of increasing period T
a/T
a/T
a/T

33. 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

34. Transition to Fourier integral (continued)
Since
It follows that
As we pass to the limit, Dw -> dw, nDw -> w so we have

35. Transition to Fourier integral (continued)
This is subject to convergence condition
Now observe that since
We have

36. Transition to Fourier integral (continued)
In the limit as T -> 
Since f(t) = 0 for t < -a/2 and t > a/2
Thus we have the Fourier transform pair for nonperiodic functions

37. Example: pulse For pulse of area 1, height a, width 1/a, we have
Note that this will have zeros at w = 2anp, n=0,+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

38. Example of pulse 5
width=1 1
-1/2
1/2
-1/10
1/10
width=0.2

39. Pulse: limiting cases
Let a -> , then f(t) -> spike of infinite height and width 1/a (delta function) -> 0
Transform -> line F(jw)=1
Thus transform of delta function contains all frequencies
Let a -> 0, then f(t) -> infinitely long pulse
Transform -> spike of height 1, width 0
Now let height remain at 1, width be 1/a
Then transform is

40. Pulse: limiting cases (continued)
Now, we are interested in limit as a -> 0 for w -> 0 and w > 0
First, consider case of small w:
So when a -> 0, 1/a -> 
As w moves slightly away from 0, it drops to zero quickly because of w/2a term in denominator (numerator <1 at all times)
So we get delta function, d(0)

41. Fourier transform of pulse width 0.1

42. Properties of delta function
Definition
Area for any g > 0
Sifting property
since

43. Some common Fourier transform pairs
Source:

44. Some Fourier transform pairs (graphical illustration)
function
transform
function
transform
Source: Physical Optics Notebook: Tutorials in Fourier Optics, Reynolds, et. al., SPIE/AIP

45. Fourier transform: Gaussian pulses

46. Properties of Fourier transforms
Simplification:
Negative t:
Scaling
Time:
Magnitude:

47. Properties of Fourier transforms (continued)
Shifting:
Time convolution:
Frequency convolution:

48. 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)

49. 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)

50. Convolution and transforms (continued)
May be visualized as
U(jw)
H(jw)
Y(jw)=H(jw)U(jw)
Input
System
Response

51. Convolution and transforms (continued)
Example
Signal is square wave, u(t)=sgn(sin(x))
This has Fourier transform
So response Y(jw) is

52. 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)

53. 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

54. 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

55. Convolution and transforms (continued)
We may assume without loss of generality that u(z) = 0 for z<0
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 t<0
Therefore h(v-t) = 0 for t > v

56. 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) = h*u
Can readily be calculated on a computer

57. Convolution: old way (graphically)

58. Convolution: old way (continued)
Source: P. S. Rha, SFSU,
ENGR449_PDFs/EE449_L5_Conv.PDF

59. Convolution and transforms (new way)
Use computer algebra programs
Some Derive scripts
Step function: u(t):=if(t<0,0,1)
Pulse of width d, amplitude a: f1(t):=if(t>=0 and t<=d,a,0)
Triangle of width d, amplitude a: triangle(t):=if(t>=0 and t<=d/2,2at/d,(if(t>d/2 and t<d,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

60. Convolution functions

61. Convolution: useful web sites

62. 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

63. 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,

64. 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
Fourier transform distinguishes these

65. 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

66. Fourier transforms of random sources (noise) (continued)

67. 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 T=sampling interval, N = number of samples, and frequency sampling interval W = 2p/NT, wk = kW

68. 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 > 2 x fmax, the highest frequency in the signal
Implies that highest frequency captured fmax < 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

69. 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

70. 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

71. Uses of DFT Magnitude Phase DFT Spectrum
DFT usage may be visualized as
Power Spectral Density
Power Spectrum
Magnitude
Phase
DFT Spectrum

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

73. 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

74. 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:

75. 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

76. Spectral leakage No discontinuities Discontinuities present
Source: National Instruments

77. 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 t/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

78. 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

79. FFT examples showing different resolution
f(x)=sin (px/5), analysis done in MATHCAD
32 sample points, T=1 sec, fs=1 resolution 1/32 Hz
64 sample points, T=1 sec, fs=1 resolution 1/64 Hz

80. 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:

81. 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

82. 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

83. 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

84. 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

85. 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

86. 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

87. 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

88. 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)/p}terms from orthonormal basis for space of functions
Any other orthonormal set of functions can also serve as basis of Fourier analysis

89. 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)

90. 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