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
Preparatory School to Winter college on 1/29/2010 Optics and Energy LAYOUT
• What is a Fourier Series? • Fourier Series Expansions – Odd and Even Functions – CiCosine andSi dSiSine Series • Fourier Series to Fourier Transforms • Fouri er Tra ns f orm s • Properties of Fourier Transform • The Convolution and Autocorrelation • The Dirac Delta Function • The Uncertainty Principle • Examples
1/29/2010 Preparatory School to Winter college on Optics and Energy 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 Preparatory School to Winter college on Optics and Energy 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 Preparatory School to Winter college on Optics and Energy 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 Preparatory School to Winter college on Optics and Energy 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
1/29/2010Preparatory School to Winter college on Optics and Energy Building a Square Wave
Add another, with an amplitude 1/5 of the original and a frequency 5i5 times th at of fh the fi rst- 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.
1/29/2010Preparatory School to Winter college on Optics and Energy 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.
1/29/2010Preparatory School to Winter college on Optics and Energy 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.
1/29/2010Preparatory School to Winter college on Optics and Energy Asymmetric Function
Some functions are neither wholly odd nor even- are asymmetric. Asymmetric functions are made up of both sine and cosine functions:
1/29/2010Preparatory School to Winter college on Optics and Energy 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 (– , ).
1/29/2010Preparatory School to Winter college on Optics and Energy The Kronecker delta function
I1 if mn J mn, K0 if mn
1/29/2010Preparatory School to Winter college on Optics and Energy Finding the coefficients, Fm, in a Fourier Cosine Series 1 Fourier Cosine Series: ft() F cos( mt ) B m m0 , ’ ’ To find Fm multiply each side by cos(m t), where m is another integer, and integrate:
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 only the m’ = m term contributes O B mmm, ' m0 ’ Dropping the from the m: Fm O ft()cos( mtdt ) a yields the coefficients for any f(t)! 1/29/2010 Preparatory School to Winter college on Optics and Energy 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 ( – , ).
1/29/2010Preparatory School to Winter college on Optics and Energy Finding the coefficients, F’m, in a Fourier Sine Series
Fourier Sine Series: 1 ft() F sin( mt ) B m m0 , ’ ’ To find Fm multiply each side by sin(m t), where m is another integer, and integrate: 1 fdf ()tmts in(') ( ')dt F si n() (mt )si n(') ( 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 O ft()sin( mtdt ') B Fmmm,'a only the m’ = m term contributes m0 Ff ()sin(tmtdt ) Dropping the ’ from the m: m O a yields the coefficients Preparatory School to Winter college on 1/29/2010 for any f(t)! Optics and Energy Fourier Series
So if f(t) is a general function, neither even nor odd, it can be writt en:
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
1/29/2010Preparatory School to Winter college on Optics and Energy 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
1/29/2010Preparatory School to Winter college on Optics and Energy 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
1/29/2010Preparatory School to Winter college on Optics and Energy 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.
1/29/2010Preparatory School to Winter college on Optics and Energy 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
1/29/2010Preparatory School to Winter college on Optics and Energy 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.
1/29/2010Preparatory School to Winter college on Optics and Energy • Generally, the Fourier transform F( ) 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
1/29/2010Preparatory School to Winter college on Optics and Energy 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 ( )
1/29/2010Preparatory School to Winter college on Optics and Energy 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: ia Fft{( a )} e 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 Preparatory School to Winter college on Optics and Energy 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 1 2()fFtedtFFt O ()it () 2
1/29/2010 Preparatory School to Winter college on Optics and Energy Properties of Fourier transform 4
6. Modulation:
1 Fft{()cos()} t [( F ) F ( )] 0002 1 Fft{()sin()}000 t [( F ) F ( )] Proof: 2 Using Euler formula, properties 1 (linearity) and 4 (frequency shifting): 1 Ff{ ()cos(ttFe )} [ { it00f ()tFe}{ itf ()t }] 0 2 1 [(F )F ( )] 2 00 Preparatory School to Winter college on Optics and Energy 1/29/2010 Scale Theorem
The Fourier transform ofldftif a scaled function, f(at): Y {(fat )} F (/)/ a a Proof: Y {(fffat )} O f (at )exp( i t ) dt
Assuming a > 0, change variables: t 0 = at Y { f ()at } O f ()exp(titadta000 [ /]) / O ft()exp([/]000 i at ) dt / a Faa(/)/ If a < 0, the limits flip when we change variables, introducing a minus sign, hence the absolute value. 1/29/2010Preparatory School to Winter college on Optics and Energy The Scale Theorem in action
f(t) F() Short pulse
t The shorter the pulse, Medium- the broader length the pulse spectrum! t
This is the essence Long of the Uncertainty pulse Principle! t
1/29/2010Preparatory School to Winter college on Optics and Energy Shift Theorem
The Fourier transform of a shifted function, ft(): a
Proof : Y ft()exp()() a iaF Y ft aO ft ( a ) exp( itdt ) Change var ia bles : tt0 a O ft (000 ) exp( i [ t a ]) dt exp(ia )O ft (000 )exp( itdt ) exp(iaF ) ( ) 1/29/2010Preparatory School to Winter college on Optics and Energy Parseval’s Theorem
Parseval’s Theorem says that the energy is the 221 same, whether you integrate over time or f ()tdt F ( ) d frequency: OO 2 2 Proof: OOf ( t ) dt f ( t ) f *( t ) dt Use ’, not , to avoid conflicts in integration variables. FVFV 11 GWGWF( exp(itd ) F *( exp( i td ) dt OOGWGW22 O HXHX FV 11 FF( ) *( )GW exp( itdtdd [ ] ) 22OOOGW HX 11 FF() *()[2 )] dd 22OO 112 FF() *() d F () d 22OO 1/29/2010Preparatory School to Winter college on Optics and Energy Parseval's Theorem in action
Time domainf(t) Frequency domain F()
t
| f(t)|2 |F()|2
t
The two areas (i.e., the light pulse energy) are the same.
1/29/2010Preparatory School to Winter college on Optics and Energy The Convolution
The convolution allows one function to smear or broaden another.
! f ()t g ()t f ()t g ()t O f (t000 )g (ttdt )
changing variables: O ft(()()– t000)() gt dt b t0 t - t0 f g fg * =
xtx
1/29/2010Preparatory School to Winter college on Optics and Energy The convolution can be performed visually: rect * rect
f ()tgt ()O fttgtdt ( –000 ) ( )
rect(t) * rect(t) = (t)
rect(x) (t)
x t
1/29/2010Preparatory School to Winter college on Optics and Energy Convolution with a delta function
fgO fttgtdt(()()– 000)() fff()tt )O f (tt000 ) ( tdt ) ft ( ) • Convolution with a delta function simply centers the function on the delta-function. • This convolution does not smear out f(t). Since a device ’s performance can usually be described as a convolution of the quantity it’s trying to measure and some instrument response, a perfect device has a delta-function instrument response.
1/29/2010Preparatory School to Winter college on Optics and Energy The Convolution Theorem
The Convolution Theorem turns a convolution into the inverse FT of thdfhFiTfhe product of the Fourier Transforms: Y {ft() gt()} =()() F G Proof: IY Y {f ( t ) gt ( )}OOJZ f ( t000 ) gt ( – t ) dt exp( i t ) dt IY K[ OOfiddf ()tgtt00JZ ( )exp (–i t )dt dt 0 K[ O ft(){(000 G exp(– it )} dt O ft(000 ) exp(– it ) dtG ( F ( G ( 1/29/2010Preparatory School to Winter college on Optics and Energy The Autocorrelation
The convolution of a function f(x))( with itself (the auto- convolution) is given by: ffO f()(t000f ttdt ) Suppose that we don’t negate any of the arguments, and we complex- conjugate the 2nd factor. Then we have the autocorrelation:
* ffO ftft()00 ( tdt ) 0
The autocorrelation plays an important role in optics.
1/29/2010Preparatory School to Winter college on Optics and Energy The Autocorrelation
As with the convolution , we can also perform the autocorrelation graphically. It’s similar to the convolution, but without the inversion.
f f fg =
x x t
Like the convolution, the autocorrelation also broadens the function in time. For real functions, the autocorrelation is syy()mmetrical (even).
1/29/2010Preparatory School to Winter college on Optics and Energy The Autocorrelation Theorem
The Fourier Transform of the autocorrelation IY2 YYJZf ()t f *(ttdx )f ()t is the spectrum! O 00 K[ Proof: IY Y JZOOOf (t00 )f *(ttdx ) exp( it )f (t 000 )f *(t t ) dt dt K[ FV* ft( )GW exp( itft ) ( tdtdt ) OO000t’ = t HX FV* $%* OOf (tit000000 )GW exp( )f (ttdtdt ) Of (tF ) ( )exp( itdt ) HX 2 O ft()exp(000 it ) dtF *() FF() *() F () 1/29/2010Preparatory School to Winter college on Optics and Energy The Autocorrelation Theorem in action
Y rect(t ) sinc( / 2) rect(t) sinc( / 2) 1 1
sinc( / 2) rect(tt ) rect( ) t sinc( / 2) ()t -1 0 1 0 2 ()t sinc2 ( / 2) sinc ( / 2) 1 1
-1 0 1 t 0 Y (t ) sinc2 ( / 2)
1/29/2010Preparatory School to Winter college on Optics and Energy The Modulation Theorem:
The Fourier Transform of E(t) cos(w0 t)
Y Et( )cos(00 t )O Et ( )cos( t ) exp( i t ) dt 1 E()titititdtFVHXex p( )ex p( )ex p( ) 2 O 00 11 E(t )exp( i [ ] tdt ) Et ( )exp( i [ ] tdt ) 22OO00 11 Y Et()cos( t ) E ( ) E ( ) 0022 0
Et()cos( t ) Y Et()cos( t ) Example: 0 0 E(t) = exp(-t2) t -0 ' 0 1/29/2010Preparatory School to Winter college on Optics and Energy The Fourier Transform of 1
• Using integration, the Fourier transform of 1 is
it i i FVeee F{1} 1edtit GW ? O HXii
• At first, we may conclude that 1 has no Fourier transform, but in fact, it can be found using the principle of duality!
1/29/2010Preparatory School to Winter college on Optics and Energy *Note that e i is neither 0 nor since ei i cos sin ?? where cos and sin do not converge and |cos | 1 and |sin | 1. ()ai But e equal to 0 since eeee ()ai i00 i with condition a is real and a > 0.
1/29/2010Preparatory School to Winter college on Optics and Energy The Spectrum
We define the spectrum, S(), of a wave E(t) to be:
2 SEt() Y {()}
This is the measure of the frequencies present in a light wave .
1/29/2010Preparatory School to Winter college on Optics and Energy The Dirac delta function
Unlike the Kronecker delta -function, which is a function of two integers, the Dirac delta function is a function of a real variable, t.
(t) Iif t 0 ()t J K0if0 if t 0 t
1/29/2010 Preparatory School to Winter college on Optics and Energy Dirac function Properties
O ()tdt 1 (t)
OO()()taftdt ()()() tafadtfa t
O exp(itdt ) 2 ( O exp[ itdt ( ) ] 2 (
1/29/2010Preparatory School to Winter college on Optics and Energy Example
Using the definition, find the Fourier transform of (t). Then deduce the Fourier transform of 1.
1/29/2010 Preparatory School to Winter college on Optics and Energy Solution
*Recall the sifting property: O (t a) f (t)dt f (a)
The F ouri er transf orm of f (t) = (t) is
Fttedtee()F {()} O () it i00 1
Then, usinggypp the duality principle, the Fourier transform of 1 is F{1} 2f ( ) 2 ( ) 2 ()
since ( ) . ()
1/29/2010Preparatory School to Winter college on Optics and Energy The Fourier Transform of (t) is 1
O (titdti ) exp( ) exp( [0]) 1 (t) )
0 t
The Fourier Transform of 1 is *(): O 1exp( itdt ) 2 (
) *()
t 0
1/29/2010Preparatory School to Winter college on Optics and Energy The Pulse Width
There are many definitions of the "w idth" or “l ength” of a wave or pul se. t
The effective width is the width of a rectangle t whose height and area are the same as those of the pulse.
Effective width Area / height: f(0) 1 (Abs value is teff tftdteff O () unnecessary f (0) fitit)for intensity.)
Advantage: It’s easy to understand. 0 t Disadvantages: The Abs value is inconvenient . We must integrate to ± . 1/29/2010Preparatory School to Winter college on Optics and Energy The Uncertainty Principle
The Uncertainty Principle says that the product of a function's widths ihiin the time d omai n ( t) andhd the frequency doma in ( ) hiihas a minimum.
Define the widths 11 assuming f(t) and tftdtFdOO( ) ( ) fF(0) (0) F() peak at 0: 11F (0) tOO ftdt() ft ()exp( i [0]) tdt ff(0) (0) f (0)
11 2(0) f OOFd() F ()exp( i $'% d FF(0) (0) F (0) (Different definitions of the widths and Combining results: the Fourier Transform yield different fF(0) (0) t 2 constants.) Ff(0) (0) or: t 2 , t 1
1/29/2010Preparatory School to Winter college on Optics and Energy Some common Fourier transform pairs
Source: http://mathworld.wolfram.com/FourierTransform.html
1/29/2010Preparatory School to Winter college on Optics and Energy Example: the Fourier Transform of a rectangle function: rect(t)
1/2 1 F ()( ) ex p(itdt ) [ ex p( it )]1/2 O 1/2 1/2 i 1 -*%[exp(ii / 2) exp( i ) -*exp(ii / 2) exp( F() -* 2i -* si(in( -* Imaginary Component = 0 F(sinc( -*
1/29/2010 Preparatory School to Winter college on Optics and Energy Example: the Fourier Transform of a GiGaussian, exp(-at2), i s it self!
Y {exp( at22 )}O exp( at )exp( i t ) dt 2 exp( / 4a ) The details are a HW problem!
exp( at 2 ) exp( 2 / 4a )
0 t 0
1/29/2010Preparatory School to Winter college on Optics and Energy The Fourier transform of exp(i 0 t)
Y exp(it00 ) O exp( it ) exp( itdt ) exp(itdt [ ] ) O 0 2(0 )
exp(i t) 0 Y {exp(i 0t)} Im t ' Re t ' ' '
The function exp(i 0t) is the essential component of Fourier analysis. It is a pure frequency. 1/29/2010Preparatory School to Winter college on Optics and Energy The Fourier transform of cos(' t)
Y cos()00ttitdtO cos ()p()()exp( ) 1 $%exp(it ) exp( it ) exp( itdt ) 2 O 00 11 exp(i [ ] t ) dt exp( i [ ] t ) dt 22OO00 ()00 ()
cos( t) Y 0 {cos(0t )}
t ' ' ' '
1/29/2010Preparatory School to Winter college on Optics and Energy Fourier Transform with respect to space
• If f(x))p, is a function of position, Fk() f ()exp( x ikxdx ) O
Y {f(x)} = F(k) x
• WftWe refer to k as the spatia llf frequency. k • Everything we’ve said about Fourier transforms bet ween the t and domains al so a ppli es t o th e x and k domain s.
1/29/2010Preparatory School to Winter college on Optics and Energy The 2D Fourier Transform
Y (2) {f(x,y)} = F(kx,ky) f(x,y) OO = f(x,y) exp[-i(kxx+kyy)] dx dy y x
If f(x,y) = fx(x) fy(y), Y (2){f(x,y)} then the 2D FT splits into two 1D FT's.
But this doesn’t always happen.
1/29/2010Preparatory School to Winter college on Optics and Energy Fourier Optics Layout
• Fourier Optics • Wave Equations and Spectrum • Separation of Variables • The Superposition Integral • Fourier Transform Pairs • Optical Systems: General Overview • Abbe Sine Condition • 2D Convolution against Impulse Response Function • Applications of Fourier Optics
1/29/2010Preparatory School to Winter college on Optics and Energy Fourier Optics
• Fourier optics is the study of classical optics using techniques involving Fourier transforms and can be seen as an extension of the Huygens-Fresnel principle. • Any widewavewhich moves fdforward can actlltually be thought of as an infinite amount of wave points, all of which could move relatively independently of each other. The theorem basically says square objects can be made by combining an infinite amount of curved objects. • If a wave is far enough away from something that it can be simplified to a square block moving forward, a Fraunhofer diffraction would be created.
1/29/2010Preparatory School to Winter college on Optics and Energy Fourier Optics 1
• When the wave is close enough than more attention must be paid to the individual wave points and the wave can only be simplified to a round ball instead of a square block, a Fresnel diffraction would be created. • Fourier optics forms much of the theory behind image processing techniques, as well as finding applications where information needs to be extracted from optical sources such as in quantum optics.
• Fourier optics makes use of the spatial frequency domain (kx, ky) as the conjugate of the spatial (x,y) domain.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Foundations of Scalar Diffraction Theory • Diffraction plays an important role in the branches of physics and engineering that deals with wave propagation. • To fully understand the properties of optical imaging and data processing system- it is essential that diffraction and its limitations on system performance be appreciated. • Firststeptoscalardiffractiontheoryis-Maxwell’s Equations
Preparatory School to Winter college on 1/29/2010 Optics and Energy From Vector to Scalar Theory
• The Maxwell’s eqqpuations in free space are written as ./ E 0 .H 0 H .E 2 t E .H / t
Preparatory School to Winter college on 1/29/2010 Optics and Energy From Vector to Scalar Theory
• Where and μ are permittivity and permeability of the medium inwhich wave is propagating. • We assume that wave is propagating in a linear, homogenous, isotropic and non-dispersive dielectric medium. • Isotropic –properties are independent of direction of polarization of the wave. • Homogenous-permittivity is constant throughout the region of propagation. • Non-dispersive - permittivity is independent of wavelength over the wavelength region occupied by the propagating wave. • Non-magnetic-magnetic permeability is always equal to free space permeability.
Preparatory School to Winter college on 1/29/2010 Optics and Energy From Vector to Scalar Theory Electromagnetic Wave Equation • Electric and magnetic fields satisfy wave equations in free space 2 .2 2 / E E o o 2 0 t 2B .2B 2 / 0 o o t2 Preparatory School to Winter college on 1/29/2010 Optics and Energy From Vector to Scalar Theory
• Since the vector wave equation is obeyed by both E and B – an identical scalar wave equation is obeyed by all the components of those vectors. • For example x-component of E obeys the equation 2E . 2E 2/ x 0 xoot2 It is possible to summarize the behavior of all components of E and B through a single scalar wave equation
Preparatory School to Winter college on 1/29/2010 Optics and Energy Propagation of Light
An optical field (light) can be described as a waveform propagating through free space (vacuum) or a material medium (such as air or glass) - the amplitude of the wave is represented by a scalar wave function u that depends on both space and time. i.e.
u= u((,r, t) Where r = (x, y, z) represents position in three dimensional space, and t represents time.
1/29/2010Preparatory School to Winter college on Optics and Energy Scalar Wave Equation
• In a dielectric medium that is, linear, isotropic , homogg,eneous, and non-dispersive- all components of the electric and magnetic field behaves identically and described by single scalar wave equation. CS1 2 DT. 2 urt(,) 0 EUct22
• whereurt(,) - represents any of the scalar field components in free space. • Fourier optics begins with scalar wave equation
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Helmholtz Equation
• One possible solution of scalar wave equation for monochromatic fields can be urt(,) ure ()it where ur() are ()ir: ()
• By substituting this in the wave equation- the time-independent form of the wave equation may be derived- known as the Helmholtz equation.
.22kur () 0
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Helmholtz Equation 1
• Where 2 k c • is the wave number,i is the imaginary; unit andur() is the time-independent- complex valued component of the propagating wave. • The propagation constant k, and the frequency are linearly related to one another, a typical characteristic of transverse electromagnetic (TEM) waves.
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Paraxial Wave Equation
• An elementaryyq solution to the Helmholtz equation takes the form ik. r ur() u0 () ke
Where kkxkxyzˆˆy kzˆ is the wave vector and kk k222 k k xyzc is the wave number.
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Paraxial Wave Equation 1
• Using the paraxial approximation, it is assumed that 22 2 kkx y kz Or equivalently sin<< where< is the angle between the wave vectork and the z-axis. As a result < < 2 kkz cos k(1 ) and 2 ikx() ky ikz< 2 xy 2 ikz ur() u0 () ke e e
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Paraxial Wave Equation 2
• Substituting u()r into the Helmholtz equation - the Paraxial wave equation is given by u ..2uik 200 T 0 z where 22 .2 T xy22 Is the transverse Laplacian operator in cartesian coordinates.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Diffraction
Diffraction: The bending of light around the edges or some obstacle is called diffraction.
Two types of diffractions 1. Fresnel diffraction- Near Field diffraction 2. FhffFraunhoffer Diffrac tion- Far field diffracti on
Huygens’ s Principle: Every point on a wave front can be considered as a point source for a spherical wave
Preparatory School to Winter college on 1/29/2010 Optics and Energy Fresnel Diffraction • Fresnel diffraction- is a process of diffraction that occurs when wave passes through an aperture and diffracts in the near field. • Any diffraction pattern observed is different in size and shape- depending on the distance between the aperture and observation plane. • It occurs due to the short distance in which the diffracted waves propagates- results in a Fresnel number greater than 1 a 2 F 1 ; L
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Fraunhofer Diffraction
• Fraunhofer diffraction- is a form of wave diffraction that occurs when field waves are passed through an aperture or slit causing only the size of an observed aperture image to change. • It is due to the far field location of observation and increasingly planar nature of outgoing diffracted waves passing through the aperture.
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Fraunhofer Diffraction • If a light source and observation screen are effect ivel y far enough from a diffracti on aperture/li/slit- the wave fronts arriving at the aperture and the screen can be considered to be collimated or planar.
• Fraunhofer diffraction occurs when the Fresnel number is less than 1.
a 2 F 1 ; L
Preparatory School to Winter college on 1/29/2010 Optics and Energy Comparison of Fresnel and Fraunhofer Diffraction
Preparatory School to Winter college on 1/29/2010 Optics and Energy Fraunhofer Diffraction Using lenses
• Using a point like source for light and collimating lens it is possible to make parallel light.
• This light will then be passed through the slit.
• AthAnother lens will focus the parallel light on obtibservation pllane.
• The same setup with multiple slits can also be used- creating a different diffraction pattern.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Fraunhofer Diffraction Using lenses
Preparatory School to Winter college on 1/29/2010 Optics and Energy Huygens’ s Principle
• Huygens’ s Principle: Every point on a wave front can be considered as a point source for a spherical wave
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Huygens-Fresnel Principle
The Huygens-Fresnel Principle describes the value of field U(P0) as a superposition of diverging spherical waves originating from the secondary sources located at each and every point P1 with in the aperture . ; exp(ikr ) 1 01 < UP()01OO UP () cos ds irs 01
Where is the angle between element of area and displacement vector between two point
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Huygens-Fresnel Principle
• According to the Huygens- Fresnel Principle every point in the plane (x’,y’,0) will act as a point source for spherical wave of type
A Ur() eikr r
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Fresnel Diffraction Integral
• The field diffraction pattern at a point ( xyzx,y,z)is) is given by
zeikr Uxyz(, ,) OO Ux (',',0) y dxdy ' ' ir; where rxxyyz (')(')222
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Fresnel Approximation
• Under (')(')xx 22 yy rz 2z The Fresnel diffraction integral can be written as
ikz ik FV 22 e HX(')(')xx yy UUU ()(xyz,,)(' OOU (''0)xy, ', 0) e2 z dddx '''dy ' iz; It shows that the propagating field is a spherical wave- orginating at the aperature and moving along z-axis. This integral modulates the amplitude and phase of the spherical wave.
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Fraunhofer Diffraction Integral
• In scalar diffraction theory, the Fraunhofer approximation is a far field approximation made to the Fresnel diffraction integral.
• UdUnder thisconditi on the quadidratic phhase factor under the integral sign is approximately unit over theentire aperture.
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Fraunhofer Diffraction Integral
The F raunh of er diff racti on occurs if kx('22 y ') z Using this approximation in Fresnel diffraction integral we get ik ()x22y ikz 2 ee2z ixxyy(' ') Uxy(, ) OO Ux (',') yekz dxdy ' ' iz;
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Plane Wave Spectrum
• The plane wave spppectrum concept is the basic foundation of Fourier Optics. • The plane wave spectrum is a continuous spectrum of uniform pllane waves and th ere i s one pl ane wave component i n th e spectrum for every tangent point on the far-field phase front. • Theea amp litude tudeo of t hat atpa plane ewaveco wave compo poenent twoudbete would be the amplitude of the optical field at that tangent point. • In the far field defined as 2 D 2 Range = ; • D is the maximum linear extent of the optical sources and is the wavelength .
1/29/2010Preparatory School to Winter college on Optics and Energy 1/29/201
The Plane Wave Spectrum 1 0
• The plane wave spectrum is often regarded as being discrete for certain types of periodic gratings. Prepar
• In reality- the spectra from gratings are continuous asa tory School to well -since no physical device can have the infinite
extent required to produce a true line spectrum. and E W nergy • For optical systems- bandwidth is a measure of how farinter college o a plane wave is tilted away from the optic axis. n • This type of bandwidth is often referred to as angular Optics bandwidth or spatial bandwidth. The Plane Wave Spectrum 2 • The plane wave spectrum arises as the solution of the homogeneous elilectromagneticwaveequation in rectangular coordinates. • In the frequency didomain, the homogeneous electromagnetic wave equation or the Helmholtz equation takes the form .22 EkErr0
2 k where r= (x, y, z) and; is the wave number of the medium.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Separation of Variables
• Solution to the homogeneous wave equation-in rectangular coordinates can be found by using the principle of separation of variables for partial differential equations. • This priilinciplesaysthat in separableorthogonal coordinat es, an elementary product solution may be constructed to this wave equation of the following form Erxyz(,x y ,)z f ()x fy () f ()z • i.e. a solution which is expressed as the product of a function of x times a function of y times a function of z.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Separation of Variables 1
• Putting this elementary product solution into the wave equation - using the scalar Laplacian in rectangular coordinates
222E EE .2 E rrr r x222yz we obtained '' '' '' 2 ffffxyz()x f ()y f ()z f fff xyz ()x f ()y f ()z f fff xyz ()x f ()y f ()z kf kfff xyz ()x f ()y f ()z 0 Rearranging fx'' () fy'' () fz'' () x y z k 2 0 fxxyz() fy () fz ()
Preparatory School to Winter college on 1/29/2010 Optics and Energy Separation of Variables 2
ff, • Three ordinary differential equations for the x y and f z, along with one separabl e condi ti on are d 2 fx() kfx2 () 0 dx2 x xx d 2 fy() kfy2 () 0 dy2 yyy d 2 fz() kfz2 () 0 dz2 zzz where 2222 kkkkxyz
Preparatory School to Winter college on 1/29/2010 Optics and Energy Separation of Variables 3
• Each of these differential equations has the same solution- a complex exponential- so that the elementary product solution
forEr is ikx()() ky kz ikx ky xyyyzxy ikz z Exyzr ()( ,,) e e e
iz k222 k k eeikx()xy ky xy • This represents a propagating or exponentially decaying uniform plane wave solution to the homogeneous wave equation. • The - sign is used for a wave propagating or decaying in the +z direction and the + sign is used for a wave propagating or decaying in the -z direction.
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Superposition Integral
• A general solution to the homogeneous electromagnetic wave equation in rectangular coordinates is formed as a weighted superposition of all possible elementary plane wave solutions as iz k222 k k ikx()xy ky xy Errxy(,xy )OO E ( k , k ) e e dkdk xy
• This ilintegral extend from minus ifiiinfinity to ifiiinfinity. • This plane wave spectrum representation of the electro- magnetic field is the basic foundation of Fourier Optics. • When z=0, the equation above simply becomes a Fourier transform (FT) relationship between the field and its plane wave content.
Preparatory School to Winter college on 1/29/2010 Optics and Energy The Superposition Integral
• All spatial dependence of the individual plane wave components is described explicitly via the exponential functions.
• The coefficients of the exponentials are only functions of spatia l wave-nuubesmbers kx aadnd ky just as in oodrdin ayary Fourie r analysis and Fourier transforms.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Free Space as a Low-pass Filter
• If 22 2 kkx yz k • Then the plane waves are evanescent (decaying)- so that any spatial freqqyuency contentinanobject plane transppyarency which is finer than one wavelength will not be transferred over to the image plane- simply because the plane waves corresponding to that content cannot pppgropagate. • In connection with lithography of electronic components, this phenomenon is known as the diffraction limit and is the reason why light of progressively higher frequency (smaller wavelength) is required for etching progressively finer features in integrated circuits.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Fourier Transform Pairs
• The Analysis equation is
ikx()xy ky f (,kkxy )OO fxye (,) dxdy • The Synthesis equation is
1 ikx() ky f (,xy ) Fk ( , k ) exy dkdk 2 OO xy x y 4
1 • The normalizing factor of4 2 is present whenever angular frequency (radians) is used, but not when ordinary frequency (cycles) is used.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Optical systems: General Overview
• An optical system is consists of an input plane and output plane. • A set of components that transforms the image f formed at the input into a different image g formed at the output. • Theoutttput image isreltdlated to the itinput image by convolilving the input image with the optical impulse response h - known as the, point-spread function - for focused optical systems. • The impulse response uniquely defines the input-output behavior of the optical system. • Theopticaxisof thesystemistaken as the z-axis. Asaresult, the two images and the impulse response are all functions of the transverse coordinates - x and y.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Optical systems: General overview 1
• Optical systems typically fall into one of two different categories. • The first is the ordinary focused optical imaging system- wherein the input plane is called the object plane and the output plane is called the image plane. • The field in the image plane is desired to be a high-quality reproduction of the field in the object plane. • The impulse response of the optical system is desired to approximate a 2D delta function- at the same location in the output plane corresponding to the location of the impulse in the input plane.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Optical systems: General overview 2
• The second typeistheoptical image ppgrocessing system- in which a significant feature in the input plane field is to be located and isolated.
• The impulse response of the system is desired to be a close replica of that feature which is being searched for in the input plane field- so that a convolution of the ftfeature agaitinst the itinput pllane fie ld will produce a bibright spot.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Input Plane and Output Plane
• The input plane is defined as the locus of all points such that z = 0. The input image f is therefore
fxy(, ) Uxyz (, ,) z0
• The output plane is defined as the locus of all points such that z = d. The output image g is therefore
gxy(, ) Uxyz (, ,) zd
Preparatory School to Winter college on 1/29/2010 Optics and Energy The convolution of input function agaitiinst impul se response f uncti on
The convolution of input function with input response function is
g(,x y ) hx (,yf ) (,x y )
gxy(, ) OO hx ( x'''''' , y y ) f ( x , ydxdy )
• The integral below tacitly assumes that the impulse response is not a function of the position (x', yy)') of the impulse of light in the input plane. • This property is known as shift invariance.
Preparatory School to Winter college on 1/29/2010 Optics and Energy The convolution of input function agaiiinst impul se response f uncti on
This equation assumes unit magnification. If magnification is present then above eqn. becomes
'''''' gxy(, )OO hM ( x Mxy , Myf ) ( x , ydxdy ) • It translates the impulse response function from x’ to x=Mx’. The relation between magnified and unmagnified response function is
hh x , y M MM
Preparatory School to Winter college on 1/29/2010 Optics and Energy Derivation of the convolution equation
• The convolution representation of the system response requires representing the input signal as a weighted superposition over a train of impulse functions by using the shifting property of Dirac delta functions. f ()tttftdt O (''' ) ( )
• The system under consideration is linear- that is the output of the system due to two different inputs is the sum of the individual outputs of the system to the two inputs - when introduced individually.
Preparatory School to Winter college on 1/29/2010 Optics and Energy Derivation of the convolution equation 1
• The outppyut of the system is then sim plified to a sin gle delta function input- which would be the impulse response of the system h(t - t'). Thus, the output of the linear system to a general input function f(t)is) is Output() t O h ( t t''' ) f ( t ) dt • The convolution equation is useful because it is often much easier to find the response of a system to a delta function input -andhd then per form th e convol uti on ab ove to fi fidhnd the response to an arbitrary input - than find the response to the arbitrary inppyut directly.
1/29/2010Preparatory School to Winter college on Optics and Energy System Transfer Function
• The Fourier transform of above equation becomes
Output() H ()() F where • Output() is the spectrum of the output signal. • H () is the system transfer function . • F() is the spectrum of the input signal. Thus, the input-plane plane wave spectrum is transformed into the output-plane plane wave spectrum through the multiplicative action of the system transfer function.
1/29/2010Preparatory School to Winter college on Optics and Energy Applications of Fourier optics
• Fourier optics is used in the field of optical information processing. • The Fourier transform properties of a lens provide numerous applications in optical signal processing such as spatial filtering, optical correlation and computer generated holograms . • Fourier optical theory is used in interferometry, optical tweezers, atom traps, and quantum computing. • Concepts of Fourier optics are used to reconstruct the phase of light intensity in the spatial frequency plane
1/29/2010 Preparatory School to Winter college on Optics and Energy References
INTRODUCTION TO FOURIER OPTICS BY JOSEPH W. GOODMAN
OPTICS BY EUGENE HECHT
Preparatory School to Winter college on 1/29/2010 Optics and Energy THANK YOU
Preparatory School to Winter college on 1/29/2010 Optics and Energy