2130-5 Preparatory School to the Winter College on Optics and Energy 1 - 5 February 2010 FUNDAMENTALS OF FOURIER OPTICS I. Ashraf Zahid Quaid-I-Azam University Pakistan FUNDAMENTALS OF FOURIER OPTICS IMRANA ASHRAF ZAHID QUAID-I-AZAM UNIVERSITY ISLAMABAD PAKISTAN PreparatorySchooltoWintercollegeon 1/29/2010 OpticsandEnergy LAYOUT • What is a Fourier Series? • Fourier Series Expansions – Odd and Even Functions – CiCosine an dSiSidSine Series • Fourier Series to Fourier Transforms • Fourie r Tra ns fo rms • Properties of Fourier Transform • The Convolution and Autocorrelation • The Dirac Delta Function • The Uncertainty Principle • Examples 1/29/2010 PreparatorySchooltoWintercollegeonOpticsandEnergy What is a Fourier Series? • A Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines (or complex exponentials). OR • The theory of Fourier series lies in the idea that most signals can be repr esen ted as a sum of sseine waves-inccudluding square waves and triangle waves-they're possibly the most-used examples. 1/29/2010 PreparatorySchooltoWintercollegeonOpticsandEnergy Sine Waves • Sine waves have lots of interesting properties- many natural operations deal with a set of differing frequency sine waves • Fourier series give a great picture of the kind of content of a signal. • A sharp transition in data generally results from a high- frequency sine wave- only high-frequency sine waves have the fast-changing edge required. • By cutting out the low frequencies- one can pick out the edges. This is particularly useful in image processing. 1/29/2010 PreparatorySchooltoWintercollegeonOpticsandEnergy Anharmonic Waves are Sums of Sinusoids Consider the sum of two sine waves (i.e., harmonic waves) of different frequencies: The resultinggp, wave is periodic, but not harmonic. Essentially all waves are anharmonic. 1/29/2010 PreparatorySchooltoWintercollegeonOpticsandEnergy Building a Square Wave Let's see how a square wave is built up- start with a sine wave: • Add another, with an amplitude 1/3 of the original and a frequency 3 times that of the first- 3rd harmonic 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Building a Square Wave Add another, with an amplitude 1/5 of the original and a frequency 5i5 times th at o fhf the first- 5th harmonic • If we carry on until the 15th harmonic- we should see a pattern emerging • It looks quite noisy- but bears resemblance to the square wave. • If we add more and more harmonics,- we get closer and closer to a square wave. 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Fourier Series Expansions • Odd and Even Functions: An important point is the difference between odd and even functions. For odd functions- on the other side of the y-axis- the function is inverted. sin is an odd function. So, we can see that an odd function is made up of sin functions onlyyy. And any combination of sin functions will produce odd functions. 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Even and Odd Functions • Even Function- cosine is an even function-mirrored about the y-axis- and so combinations of cosine functions produce even functions. 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Asymmetric Function Some functions are neither wholly odd nor even- are asymmetric. Asymmetric functions are made up of both sine and cosine functions: 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Fourier Cosine Series Because cos(mt) is an even function (for all m), we can write an even function, f(t), as: 1 f(t) B Fm cos(mt) m 0 where the set {Fm; m = 0, 1, … } is a set of coefficients that define the series. And where we’ll only worry about the function f(t) over the interval (–444,4). 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy The Kronecker delta function I1 if mn J mn, K0 if mn 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Finding the coefficients, Fm, in a Fourier Cosine Series 1 FourierCosineSeries: ft() F cos( mt ) B m m0 , ’ ’ TofindFm multiplyeachsidebycos(m t),wherem isanotherinteger,andintegrate: 1 ft()cos(( ) cos( mtdt' )cos()cos( ) F cos( mt ) cos( mtdt' ) OO B m m0 I if m m ' But: cos()mtcos( m' t ) dt J O K0'if mm mm, ' 1 So: ft()cos( mtdt ') F a onlythem’ = m termcontributes O B mmm, ' m0 ’ Droppingthe fromthem: Fm O ft()cos( mtdt ) a yields the coefficientsfor anyf(t)! 1/29/2010 PreparatorySchooltoWintercollegeonOpticsandEnergy Fourier Sine Series Because sin(mt))( is an odd function (for all m), we can write any odd function, f(t), as: 1 f (t) F sin(mt) B m m 0 where the set {F’m; m = 0, 1, … } is a set of coefficients that define the series. where we’ll only worry about the function f(t) over the interval ( –,). 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Finding the coefficients, F’m, in a Fourier Sine Series FourierSineSeries: 1 ft() F sin( mt ) B m m0 , ’ ’ TofindFm multiplyeachsidebysin(m t),wherem isanotherinteger,andintegrate: 1 fdf ()tmts in(') ( ')dt F s in() (mt )s in ((') m ') t d t OO B m m0 But: I if mm ' sin(mt ) sin( m ' t ) dt J O K0'if m m mm,' So: 1 ft()sin( mtdt ') F O B mmm,'a onlythem’ = m termcontributes m0 Ff ()sin(tmtdt ) Droppingthe’ fromthem: m O a yieldsthecoefficients PreparatorySchooltoWintercollegeon 1/29/2010 foranyf(t)! OpticsandEnergy Fourier Series So if f(t) is a general function, neither even nor odd, it can be wr itten: 11 ft()BB F cos( mt ) F sin( mt ) mm mm00 even comppponent odd component where F f (t) cos(mt) dt and F f (t) sin(mt) dt m O m O 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Fourier Series to Fourier Transform • Periodic signals- represented by linear combinations of harmonically related complex exponentials • To extend this to non-periodic signals, we need to consider a periodic signals with infinite period. • As the period becomes infinite, the corresponding frequency components form a continuum and the Fourier series sum becomes an integral • FiFourier tftransform - isacomplex valdlued ftifunction inthe frequency domain 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Comparison between Fourier series and Fouri er trans f orm Fourier series Fourier transform • Support periodic function • Support non-periodic fifunction • Discrete frequency • Continuous frequency spectrum spectrum 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy The Fourier Transform Consider the Fourier coefficients. Let’s define a function F(m) that incorporates both cosine and sine series coefficients, with the sine series distinguished by making it the imaginary component: F m F iF’ ( ) m – m = O f ()cos(tmtdt ) iftO ()sin( mtdt ) Let’s now allow f(t) to range from to , so we’ll have to integrate from to , and let’s redefine m to be the “frequency,” which we’ll now call : Fftitdt() ()exp( ) The Fourier O Transform F( ) is called the Fourier Transform of f(t). It contains equivalent information to that in f(t). We say that f(t) lives in the time domain, and F( ) lives in the frequency domain. F( ) is just another way of looking at a function or wave. 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy The Inverse Fourier Transform The Fourier Transform takes us from f(t)to) to F( ). Recall our formula for the Fourier Series of f(t) : 11 f ()tFmtFmt cos( )' sin( ) BBmm mm00 Now transform the sums to integrals from to , and again replace Fm with F( ). Remembering the fact that we introduced a factor of i and included a factor of 2, we have: 1 ft( ) F ( ) exp( itd ) Inverse 2 O Fourier Transform 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy The Fourier Transform and its Inverse The Fourier Transform and its Inverse: Fftitdt( ) O ( ) exp( ) FourierTransform 1 ft() F ( )exp( i td ) 2 O Inverse Fourier Transform So we can transform to the frequency domain and back. Interestingly, these transformations are very similar . There are different definitions of these transforms. The 2 can occur in several places, but the idea is generally the same. 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy • Generally, the Fourier transform F(2) exists when the Fourier integral converges • A condition for a function f(t) to have a Fourier transform is, f(t) can be completely interable. • This condition is sufficient but not necessary O f (t) dt 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Fourier Transform Notation There are several ways to denote the Fourier transform of a function. If the function is labeled by a lower -case letter, such as f, we can write: f(t) ® F() If the function is already labeled by an upper-case letter, such as E, we can write: Et() Y { Et()} or: Et() E ( ) 1/29/2010PreparatorySchooltoWintercollegeonOpticsandEnergy Properties of Fourier transform 1 Linearity: For any constants a, b the following equality holds: Faft{ ( ) bgt ( )} aFft { ( )} bFgt { ( )} aF ( ) bG ( ) 2 Scaling: For any constant a, the following equality holds: 1 Ffat{( )} F ( ) ||aa 1/29/2010 Preparatory School to Winter college on Optics and Energy Properties of Fourier transform 2 3 Time shifting: Fft{( a )} e ia F () Proof: iii t i a it0 Fft{( a )}OO ft ( ae ) dte fte ()00 dt • Frequency shifting: it0 Fe{ ft()} F()0 Proof: it00 it it Fe{()} ftO e fte () dt F (0 ) 1/29/2010 PreparatorySchooltoWintercollegeonOpticsandEnergy Properties of Fourier transform 3 5. Symmetry (Duality): FFt{()}2 f ( ) Proof: The inverse FiFourier transform is 1 fff()tF1 {f ( )}Fed ( ) it O 2 therefore