Chapter 9 Fourier Optics

Chapter 9

Fourier Optics Part I

Fourier Transform Fourier Series

• Periodic function f(t) = f(t+T) for all t – T is the period

– Period related to frequency by T = 1/푓0 = 2휋/휔0

– 0 is called the fundamental frequency

• So we have  f (t)  a0  (an cos2nf0  bn sin 2nf0 ) n1   a0  (an cosn0  bn sin n0 ) n1

• n0 = 2n/T is nth harmonic of fundamental frequency Fourier Series

• Calculation of Fourier coefficients hinges on orthogonally of sine, cosine functions

T 0 sin(m0t)cos(n0t)dt  0, all m,n

T T  sin(m t)sin(n t)dt   0 0 0 2 mn

T T  cos(m t)cos(n t)dt   0 0 0 2 mn 1, when 푚 = 푛 where 훿 = is called Kronecker delta 푚푛 0, when 푚 ≠ 푛 Also T sin m0t dt  0, allm 0 T cosm0t dt  0, all m 0 Fourier Series

• Based on above relationship, one can calculate the Fourier coefficients as

2 T a   f (t)cos(n t) dt n T 0 0

2 T b   f (t)sin(n t) dt n T 0 0

• 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 Fourier Series • Square wave 1, 0  t 1/ 2 f (t)   1,1/ 2  t  1

1 T/2 T

-1

4  sin 30t sin 50t sin 70t  f (t)  sin 0t       3 5 7  Fourier Series • Square wave Gibbs phenomenon: ringing near discontinuity

1.5

1.0 n = 1 n = 2 0.5 n = 3 n = 4

)

t 0.0

(

f

-0.5

-1.0

-1.5 0.0 0.5 1.0 1.5 2.0 t Fourier Series

• Triangular wave +V T/2 T

-V

 4V t, 0  t  T / 4  T  4V f (t)   t  2V , T / 4  t  3T / 4  T 4V  V  t, 3T / 4  t  T  T

8V  sin 30t sin 50t sin 70t  f (t)  2 sin 0t  2  2  2    3 5 7  Fourier Series

Parseval's theorem • If some function f(t) is represented by its Fourier expansion on an interval [-l, l], then

2 l   1 2a0 2 2 f(x)anbn l  2l 4n1 n1

• Useful in calculating power associated with waveform Fourier Series

Exponential form of Fourier Series  f (t)  a0  an cosn0t  bn sin n0t n1 Since 1 cosn t  ein0t  ein0t  0 2 1 sin n t  ein0t  ein0t  0 2 j Then   ein0t  ein0t ein0t  ein0t    f (t)  a0  an  bn  n1 2 2i 

  an  ibn in0t an  ibn in0t  f (t)  a0   e  e  n1 2 2  Fourier Series

Introducing new coefficients a  ib a  ib c~  n n , c~  n n , c~  a n 2 n 2 0 0 We have  ~ ~ in0t ~ in0t f (t)  c0  cne  cne  n1 Or  ~ in0t f (t)  cne n

The coefficients can easily be evaluated

~ an  ibn 1 T i T c    f (t)cosn t dt   f (t)sin n t dt n 2 T 0 0 T 0 0

1 T 1 T in t   f (t)cosn t  isin n t dt   f (t)e 0 dt T 0 0 0 T 0 Fourier Transform 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 Fourier Transform Fourier analysis for nonperiodic functions The Fourier coefficients

~ 1 T in0t c   f (t)e dt n T 0

1 a / 2 in t   V e 0 dt T a / 2 0 Thus,

V  ein0a / 2  ein0a / 2   a  sin(n a / 2)  ~ 0   0  0  cn     V0   n  2i  2  n0a / 2  Fourier Transform Fourier analysis for nonperiodic functions

Since 푇 = 2휋/휔0, one has a sin(n a / 2) a  sin(na /T )  ~  0  cn  V0    V0   T  n0a / 2  T  na /T  We are interested in what happens as period T gets larger, with pulse width a fixed

– For graphs, a = 1, V0 = 1 Fourier Transform Fourier analysis for nonperiodic functions

0.25 0.6 a/T a/T 0.5 0.2

0.4 0.15

0.3 0.1 0.2 0.05 0.1 0

coefficient value, T=5 coefficient value, T=2 0 0 50 100 150 0 50 100 150 -0.05 -0.1 -0.1 -0.2 frequency frequency

0.12 a/T 0.1

0.08

0.06

0.04

0.02

coefficient value, T=10 0 0 50 100 150 -0.02

-0.04 frequency Fourier Transform Fourier analysis for nonperiodic functions

• We can define f(inw0) in the following manner

~ 1 a / 2 in0t c   V e dt n T a / 2 0 ~ a / 2 in0t cnT  a / 2V0e dt ~ ~ ~ cn cn F(in0 )  cnT  2  2 0 

Since difference in frequency of terms ∆휔 = 휔0 in the expansion. Hence

F(in ) c~  0 n 2 Fourier Transform Fourier analysis for nonperiodic functions • Since  ~ in0t f (t)  cne n • It follows that  F(in ) f (t)  lim  0 ein0t  T  n 2

• As we pass to the limit, ∆휔 → 푑휔, 푛∆휔 → 휔 so we have

1  it f (t)   F(i)e d 2  Fourier integral Fourier Transform • This is subject to convergence condition

 f (t) dt    • Now observe that since

~ 1 T in0t c   f (t)e dt n T 0 • We have ~ a / 2 in0t cnT  F(in0 )  a / 2 f (t)e dt • In the limit as 푇 → ∞ a / 2 it  it F(i)  a / 2 f (t)e dt   f (t)e dt Fourier Transform

• Since f(t) = 0 for t < -a/2 and t > a/2 • Thus we have the Fourier transform pair for nonperiodic functions F(i)  Ff (t) f (t)  F -1 F(i) Fourier Transform • For pulse of area 1, height a, width 1/a, we have

1/ 2a it sin / 2a F()   ae dt  1/ 2a  / 2a

• Note that this will have zeros at 휔 = 2푎푛휋, 푛 = 0, ±1, ±2, … • Considering only positive frequencies, and that “most” of the energy is in the first lobe, out to 2푎휋, we see that product of 1 bandwidth 2푎휋 and pulse width = 2휋 푎 a

t -1/2a 1/2a T Fourier Transform

5 1.5

1.0 1 Width = 1

-1/2 1/2 Width = 0.2 0.5 -1/10 1/1

F 0

0.0

-0.5 -40 -20 0 20 40  Fourier Transform • Let 푎 → ∞ , but keep the same area, then 푓(푡) →spike of infinite height (delta function) – Transform → line F(i) =1 – Thus transform of delta function contains all frequencies

sin / 2a F()  lim 1 a  / 2a

So we get delta function, f(t) = (t) Fourier Transform

Properties of delta function  (x) 1 0 x  0 • Definition  (x)   x  x  0 0   (x)dx  1 (x)dx • Area       for any  > 0

• Sifting property  (x) f (x)dx  f (0)   (x- x0)  (x  x0 ) f (x)dx  f (x0 ) 1 

x 0 x0 since 0 x  x0 (x  x0 )    x  x0 Fourier Transform Some common Fourier transform pairs F{1}  2 () F{ (x)} 1

f (x) F () f (x) F() 1 2 … x  x  0 0 0 0

f (x) F () f (x) F() 1 1 … x  x  0 0 0 0 Fourier Transform

f (x) A (k)

k x f (x)   x  d / 2  x  d / 2 0 F{ f (x)}  eid / 2  eid / 2  2cos(d / 2) f (x) B (k) f (x)   x  d / 2 x  d / 2 id / 2 id / 2 x k F{ f (x)}  e  e  2isin(d / 2) 0 f (x)  cos x 0 f (x) A (k) F{ f (x)}    (  0 )   ( 0 )

x f (x)  sin0 x k 0 F{ f (x)}  i (  0 )  ( 0 )

f (x) B (k)

x k 0 Fourier Transform Some common Fourier transform pairs

function transform function transform Fourier Transform

Properties of Fourier transforms

• Simplification:  F(i)  20 f (t)cost dt, f (t) even  F(i)  20 f (t)sint dt, f (t) odd

• Negative t: Ff (t)  F *(i)

• Scaling – Time: 1  i  Ff (at)  F  a  a  – Magnitude: Faf (t)  aF(i) Fourier Transform

Properties of Fourier transforms • Shifting: Ff (t  a)  F(i)eia

i0t Ff (t)e  Fi( 0 ) 1 Ff (t)cos t  Fi(  ) Fi(   ) 0 2 0 0 1 Ff (t)sin t  Fi(  ) Fi(   ) 0 2 0 0

• Time convolution:

1  F F1(i)F2 (i) - f1( ) f2 (t  )d

• Frequency convolution:

1  F  f (t) f (t)  F (i)F i(  )d 1 2 2  1 2 Linear System • 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

• The response of something (e.g., a circuit) to a delta function is called its “impulse response” or called “point spread function” in optics, often denoted h(t) Linear System

• The output signal is the convolution of the input signal and response

+∞ 푌 푡 = −∞ 푥 휏 푓 푡 − 휏 푑휏 Or 푌 휔 = 푥 휔 푓(휔) Convolution Convolution and transforms (continued) • Example – Signal is square wave, u(t) = sgn(sin(x))

 sin(2n 1) t u(t)   0 n1 2n1

– This has Fourier transform

 i   (2n 1)0   (2n 1)0  U (i)           2 n1  2   2 

– So response Y(i) is

 i   (2n 1)0   (2n 1)0  Y (i)  H (i)U (i)  H (i)          2 n1  2   2  Part II

Fourier Transform and Optics Diffraction Theory exp(ikr) E   dxdy Fraunhofer diffraction A  Aperture r

R  X 2 Y 2  Z 2 r  X 2  (Y  y)2  (Z  z)2  R 1 (Yy  Zz) / R2 

 exp(ikR) E(Y, Z)  A  expik(Yy  Zz) / R dydz R Aperture   E(Y , Z)  A( y, z)expik(Yy  Zz) / R dydz y Y  

kY  kY / R Let  , then P(Y,Z)Z kZ  kZ / R dydz r R Y   x i(kY ykZ z) E(kY ,kZ )  A( y, z)e dydz X   z Z

  i(kY ykZ z) E(kY ,kZ )  A(y, z)e dydz   Diffraction Theory

Fraunhofer diffraction

  i(kY ykZ z) E(kY ,kZ )  A(y, z)e dydz  

• Each image point corresponds to a spatial frequency. • The field distribution of the Fraunhofer diffraction pattern is the Y Fourier transform of the aperture y function: P(Y,Z)Z dydz r R Y E(kY ,kZ )  F{A(y, z)}. x z X Z Diffraction Theory The single slit:

A0 | z | b / 2 A(z)   0 Otherwise b/ 2 ikZ z kZ b E(kz )  A0e dz  A0bsinc b/ 2 2

A (z) E (kZ)

z kZ

-b/2 b/2

2/b Diffraction Theory Rectangular aperture:

a / 2 b/ 2 i(kY ykZ z) kY a kZ b E(kY ,kZ )  A0e dydz  A0absinc sinc a / 2 b/ 2 2 2 The double slit (with finite width):

f (z) h (z) g (z)  = z z z -b/2 b/2 -a/2 a/2 -a/2 a/2

F (kZ) H (kZ) G (kZ) = kZ k Z kZ Diffraction Theory Three slits:

 ikZ z E(kZ )  [ (z  a)  (z)  (z  a)] e dz 1 2cos kZ a 

|F (k )|2 f (z) F (kZ) Z

z kZ kZ -a 0 a Lens as a Fourier Transform System Spatial Filter

Spatial filters provide a convenient way to remove random fluctuations from the intensity profile of a laser beam. This greatly improves resolution — which is especially critical for applications like holography and optical data processing. Spatial Filter Spatial Filter 2D Lattice Diffraction 3D Lattice Diffraction