Phys 322 Chapter 11 Lecture 31 Fourier Optics

Introduction Fourier Transform: definition and properties Fourier optics

Introduction Motivation: 1) To understand how an image is formed. 2) To evaluate the quality of an optical system.

Optical system Star (Telescope) Film

Delta function Fourier transform Inverse Fourier and filtering transform

The quality of the optical system: • The image is not as sharp as the object because some high frequencies are lost. • The filtering function determines the quality of the optical system. Fourier transform: 1D case Any function can be represented as a Fourier integral: angular spatial frequency k  2 /  1  f x  Ak cos kx dk  B k sin kx dk     00 Fourier sine transform of f(x) Fourier cosine transform of f(x)   Ak   f x cos kx dx   Bk   f x sin kx dx     Alternatively, it could be represented in complex form:

1  f x   Fk eikxdk complex 2  functions!  ikx Fk   f x e dx Fk A k  iBk Fkei k   Fourier transform 1  F(k) is Fourier transform of f(x): f x   Fk eikxdk 2  Ffk  Y  x  Fk   f x e ikxdx f(x) is inverse Fourier transform of F(k): 

fx  Y -1  Fk 

Application to waves: 1   If f(t) is shape of a wave in time: f t   Feitd 2   F   f t e itdt  Reminder: the Fourier Transform of a rectangle function: rect(t) 1/2 1 Fitdtit( ) exp( ) [exp(   )]1/2  1/2 1/2 i 1 [exp(ii / 2) exp( i  exp(ii / 2) exp(    2i   sin(  F()     Imaginary F(sinc(  Component = 0

 Sinc(x) and why it's important

Sinc(x/2) is the Fourier transform of a rectangle function.

Sinc2(x/2) is the Fourier transform of a triangle function.

Sinc2(ax) is the diffraction pattern from a slit.

It just crops up everywhere... Transform of the Gaussian function

2 f x  Ceax Amplitude: C  a   e1/ 2 a  0.607 a  Standard deviation  : f()=Ce-1/2

 x  1 2a

Fourier transform:   2 Fk   f x e ikxdx  Ceax eikxdx   2 Fk  ek 4a 0.607  k  2a

For a Gaussian function Fourier transform is also a Gaussian and  x k  1 Properties of FT The Fourier Transform of a sum of two functions

f(t) F()

t 

g(t) G() Y {()af t bgt ()} t  aftbgtYY{ ( )} { ( )} F() + f(t)+g(t) G()

Also, constants factor out. t  Properties of FT : The Fourier Transform of the complex conjugate of a function

Y ft**() F ( )

Proof:  Y f **()t  f ()exp(titdt )  *   ft ( )exp( itdt )  *    f (titdt )exp( [ ] )       F * ( ) Derivative Theorem

The Fourier transform of a derivative of a function, f’(t): Y {'()}ft iF ()

Proof:  Y {f '()}tftitdt '()exp( ) 

Integrate by parts:   f ()[(ti  )exp(  itdt )] Remember that the  function must be zero  at ±∞, so the other term, [f(t) exp(-it)] +∞ -∞ iftitdt( ) exp( ) vanishes.    iF(  if t  0 The Dirac delta function  ()t   0 if t  0

It’s useful to think of the delta function as the limit of a series of peaked continuous functions.

2 fm(t) = m exp[-(mt) ]/√

(t)

f3(t)

f2(t)

f1(t)

t The (Dirac) delta function

0 x  0 (x)  x   f(x)  x  0   x dx  1 x 

For arbitrary function f(x): 0 x  x0   x  x0    x  x0  f x  x dx  f 0   f x  x  x dx    0 f x0  

Sifting property -  function can extract only one value of f Delta function: Fourier transform 1  Fourier transform of  function: f x   Fk eikxdk  2  Fk   x e ikxdx   ikx  Fk   f x e dx 

Fk  eik0 =1

 function can be represented as a sum of an infinite number of harmonic components with amplitudes 1:

   1 1 x   eikxdk   eikxdk 2  2 

 x Y -1 1 Inverse Fourier transform of unity Delta function: displacement and phase (x) What happens if  function is shifted?  Fk   x  x eikxdx   0  x0 x amplitude: 1 Fk  eikx0 phase: kx0

-1 ixk 0  xx0  Y  e 

Fourier transform of a function that is displaced in space (or time) is the transform of the undisplaced function multiplied by an exponential that is linear in phase ixk 0 YYf xx0   fxe  Delta function: inverse Fourier transform

1  f x   Fk eikxdk 2  Fk   k  k   0 Fk   f x e ikxdx  1  f x   k  k eikxdk   0 2  1 f x  eik0x 2 Superposition of delta functions   d   d  f x   x     x    2   2  FT of sum of functions is a sum of FT of individual functions:

Y fx  eidkk/2 e id /2 2coskd / 2

1  Fk  A k  iBk Fkei k  f x  Ak cos kx dk  B k sin kx dk   00 Even function: only cosines!

If function f(x) is real and even its FT is also real and even.