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**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

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**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

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**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

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**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

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**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)

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**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 n0 = 2n/T is nth harmonic of fundamental frequency

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**How to calculate Fourier coefficients**

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

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**How to calculate Fourier coefficients (continued)**

And we also need

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**How to calculate Fourier coefficients (continued)**

Step 1. integrate both sides: Therefore

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**How to calculate Fourier coefficients (continued)**

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

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**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

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**Calculation of Fourier coefficients: examples**

Square wave (in class) 1 T/2 T -1

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**Calculation of Fourier coefficients: examples (continued)**

Gibbs phenomenon: ringing near discontinuity Result Source:

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**Calculation of Fourier coefficients: examples (continued)**

Triangular wave (in class) +V T/2 T -V

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**Calculation of Fourier coefficients: examples (continued)**

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

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**Other simplifying assumptions: half-wave symmetry**

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

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**Other simplifying assumptions: half-wave symmetry**

Can be shown

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**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

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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

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**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

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**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

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**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

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**Exponential form of Fourier Series**

Previous form Recall that

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**Exponential form of Fourier Series (continued)**

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

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**Exponential form of Fourier Series (continued)**

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

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**Exponential form of Fourier Series (continued)**

The coefficients can easily be evaluated

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**Exponential form of Fourier Series (continued)**

Sometimes coefficients written in real and complex terms as where

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**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

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**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

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**Fourier analysis for nonperiodic functions (continued)**

Start with previous formula: This can be readily evaluated as

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**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

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**Effect of increasing period T**

a/T a/T a/T

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**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

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**Transition to Fourier integral (continued)**

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

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**Transition to Fourier integral (continued)**

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

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**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

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**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

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

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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

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**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)

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**Fourier transform of pulse width 0.1**

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**Properties of delta function**

Definition Area for any g > 0 Sifting property since

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**Some common Fourier transform pairs**

Source:

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**Some Fourier transform pairs (graphical illustration)**

function transform function transform Source: Physical Optics Notebook: Tutorials in Fourier Optics, Reynolds, et. al., SPIE/AIP

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**Fourier transform: Gaussian pulses**

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**Properties of Fourier transforms**

Simplification: Negative t: Scaling Time: Magnitude:

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**Properties of Fourier transforms (continued)**

Shifting: Time convolution: Frequency convolution:

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**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)

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**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)

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**Convolution and transforms (continued)**

May be visualized as U(jw) H(jw) Y(jw)=H(jw)U(jw) Input System Response

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**Convolution and transforms (continued)**

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

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**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)

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**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

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**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

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**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

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**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

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**Convolution: old way (graphically)**

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**Convolution: old way (continued)**

Source: P. S. Rha, SFSU, ENGR449_PDFs/EE449_L5_Conv.PDF

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**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

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**Convolution functions**

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**Convolution: useful web sites**

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**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

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**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,

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**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

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**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

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

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**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

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**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

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**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

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**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

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**Uses of DFT Magnitude Phase DFT Spectrum**

DFT usage may be visualized as Power Spectral Density Power Spectrum Magnitude Phase DFT Spectrum

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**Power measurements and DFT**

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

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**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

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**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:

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**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

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**Spectral leakage No discontinuities Discontinuities present**

Source: National Instruments

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**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

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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

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**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

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**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:

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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)

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**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

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Mike Doggett Staffordshire University

Mike Doggett Staffordshire University

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