Phys 322 Chapter 11 Lecture 33 Fourier Optics

Optical applications: Linear optical system Fraunhofer diffraction Other applications Optical application: linear system

Function f(y,z) passes through an optical system and is transformed into g(Y,Z): gY ,Z L f y, z The system is linear if: • multiplying f(y,z) with constant a produces output ag(Y,Z) L af y, z aL f y, z • when the input is a sum of two (or more) functions,

f1 (y,z) +f2 (y,z), the output would be g1 (y,z) +g2 (y,z), where g1 and g2 are outputs of f1 and f2, respectively

L f1y, z  f2 y, z L f1y, zL f2 y, z Optical application: linear system

gY ,Z L f y, z

Linear system is space invariant if it possesses the property of stationarity. changing the position of input merely changes the position of output without altering its functional form

Idea behind: at each point in image plane g(Y,Z) is a linear superposition of outputs arising from each of the individual points on the object f(y,z) Optical application: linear system

gY ,Z L f y, z

  f y, z   f y', z ' y' y  z'z dy'dz'      gY ,Z L   f y', z' y'y  z'z dy'dz'     gY ,Z   f y', z ' L y'y z'z dy 'dz'  Impulse response Optical application: linear system   g Y ,Z  f y', z ' L y'y z'z dy 'dz' yz  

Impulse response: gY ,Z L f y, z  L  y' y z'z

image of a single object point at (y’,z’)

If we knew the impulse response of the system, we could construct the image g of any object f. Optical application: linear system

Example: lens system Image: upright, same size

I0(y,z) - irradiance in object plane Image blur due to diffraction etc: S y, z,Y ,Z  2f point spread function 2f Image intensity due to one element radiant flux I0(y,z)dydz

dIi Y ,Z S y, z,Y ,Z I0 y, zdydz If the object emits incoherent light, intensities add up

 I Y ,Z  S y, z, Y ,Z I y, z dydz  i  0  Optical application: linear system

 I Y ,Z  S y, z, Y ,Z I y, z dydz  i  0  Meaning of S Suppose object is just a single

luminous point at y0, z0

  I Y ,Z  S y, z, Y ,Z A y  y  z  z dydz  i  0 0  A = 1 W/m2

Ii Y ,Z  AS y0, z0,Y ,Z 

Ideal system will show far-field diffraction, Airy distribution Optical application: linear system

dIi Y ,Z  AS y0, z0,Y,Z  Space invariance: Spread function is the same for

any x0,y0 (reality: spread function varies slightly with position)

for MT=1 S y, z,Y ,Z S Y  y,Z  z

For  source in center:S Y ,Z 

 I Y ,Z  I y, z S Y  y,Z  z dydz  i  o  convolution integral Linear systems summary

For simplicity, considering the case 1) incoherent light, and

2) MT = +1. The flux density arriving at the image point from dydz is

dIi (Y, Z)  S(y, z;Y, Z)I0 (y, z)dydz Point-spread function z Z

I0 (y,z) Ii (Y,Z) y Y   Ii (Y, Z)  S(y, z;Y, Z)I0 (y, z)dydz     Ii (Y, Z)  S(y, z;Y, Z)I0 (y, z)dydz   z Z Ii (Y,Z) Point-spread function I0 (y,z) Example: I (y, z)  A (y  y ) (z  z ) y Y 0 0 0   Ii (Y, Z)  A (y  y0 ) (z  z0 )S(y, z;Y, Z)dydz  

 AS(y0 , z0 ;Y, Z) The point-spread function is the irradiance produced by the system with an input point source. In the diffraction-limited case with no aberration, the point-spread function is the Airy distribution function. The image is the superposition of the point-spread function, weighted by the source radiant fluxes.

  Ii (Y, Z)  S(y, z;Y, Z)I0 (y, z)dydz z   I (y,z) Z  0 Ii (Y,Z) I0 (y, z)   Ai (y  yi ) (z  zi ) i y Y Ii (Y, Z)   Ai S(yi , zi ;Y, Z) i Transfer functions for characterization of optical systems:

  I (,Y Z ) S ( Y y , Z z ) I (,) y z dydz i   0

Iyz0 (,) Syz (,)

YYY{(,IYZi )} {(,)} I0 yz {(,)} Syz

Y {(,)}Syz Tk (YZ , k ) Optical transfer function T (OTF) M (,kk )exp(, i kk ) Modulation transfer function M (MTF) YZ YZ Phase transfer function  (PTF)

We use these functions to characterize the quality of an optical system and to reconstruct the original (unknown) object. Optical application: Fraunhofer diffraction y Y Light r P (screen) x Z z Each area dydz is a source of spherical waves:

   E y, zdydz itkr  Source strength per dE  A e yz R unit area E y, zei y,zdydz dE  A eik YyZz / R R aperture function dE A y, zeikYyZz/ Rdydz

 EY ,Z  A y, z eikYyZz / Rdydz  Optical application: Fraunhofer diffraction y Y Light r P (screen) x Substitute:

Z kY  kY / R z  k  kZ / R EY ,Z  A y, z eikYyZz / Rdxdz Z   For each point Y,Z Ek ,k  A y, z e ikY ykZ z dydz Y Z  there is corresponding  spatial frequency The field distribution in the Fraunhofer diffraction is the Fourier transform of the field distribution across the aperture (aperture function) E kkYZ,, Y T  yz

1 The inverse transform:T yz,, Y  EkYZ k  Optical application: single slit  y Y Ek ,k  A y, z e ikY ykZ z dydz Y Z  Light 

EkY ,kZ F A y, z  R k  kY / R k  kZ / R Z Y Z z

A 0 when z  b / 2 A y, z   0 when z  b / 2

 b/ 2 Ek F A z  A z eikZ zdz A eikZ zdz Z  0   b/ 2 k b Ek A bsinc z k  kZ / R  k sin Z 0 2 Z Got the same result Optical application: double slit Single A(x) A(x) Double slit slit A0 A0

E(k) E(k) Phys 322 Chapter 11 Lecture 33 Fourier Optics

Other applications: Time and frequency domains FTIR and many more Time and frequency domain

E(t)=f(t) - electric field depends on time (time domain) Energy flux ~|E(t)|2 = |f(t)|2  2 2 Total emitted energy:   f t dt f t  f tf *t 

F()=F {f(t)} Fourier transform (frequency domain) |F()|2 = measure of energy per unit frequency interval          2  1 * it  1 *  it   f t dt  f t  F e ddt  F  f t e dtd   2   2   

  2 2 1  2 |F()| - f t dt  Fd  2  power spectrum or   Spectral energy Parseval’s formula distribution Emission spectrum of excited atoms/molecules Emission intensity is proportional to the number h h of molecules in excited state N  dN  N N  N e t  N et / dt 0 0

t / rate of transition Emission intensity: I  I0e for t0 (damping constant)  t / 2  = 1/ = lifetime  Et E0e cos0t  F E et / 2 cos t eitdt  0 0 0   2 2 2 E  / 4 0  F 2 2 2  0   / 4 Lorentzian profile

full width at half maximum:   fwhm =  0 natural linewidth Example problem:

An excited state lifetime of a chlorophyll molecule is 5 ns. What is the natural linewidth of its emission spectrum, if transition maximum occurs at 670 nm. Express linewidth in nm

fwhm =  = 1/ = 1/(5 ns) = 2×1010 Hz     c c  2  2  2  2.81015 Hz  0 0  0 0 0   0  c c  c fwhm  2  2  2 2 0  / 2 0  / 2 0

-3 fwhm = 4.9×10 nm  0 Reality: fwhm ~ 10 nm - inhomogeneus broadening FTIR spectrometer (Fourier Transform InfraRed) IR detector What would be recorded if IR source Movable emits a single wavelength? mirror

IR source  2 I A  cos  Mirror    Acos c  What if there are two wavelengths?  N    I A cos    i i i c  i1       I A cos d   c  FTIR spectrometer: qualitative approach  IR detector     I A cos d Movable  c mirror  

IR source

Mirror Measure I()

FT gives spectrum:AI Y    FTIR spectrometer

IR detector No sample Sample Movable mirror

IR source

Mirror With sample

Transmittance spectrum of a sample (polystyrene film) FTIR formal math: Autocorrelation

IR detector Electric field at detector: Movable mirror Ed t E t  Et  

IR source     / c  Measure intensity averaged in time: Mirror  I Et 2  Et  E t   2 dt  d       I  E 2 t dt  2  E t E t  dt   E 2 t  dt        I  2  E 2 t dt  2  E t E t  dt     Autocorrelation Constant background Autocorrelation  c  f t f t  dt  ff  

2 Wiener-Khinchine theorem: Y cFff   

IR detector   Movable I  E t E t  dt  mirror 

2 IR source Y IF  

Mirror   c  f t h t  dt f h Cross-correlation: hf   Extraction of signal from noise

Sine function “hidden” in noise.

Autocorrelation of white noise is 0. Spectral grating. E(t) - electric field depends on time (time domain)

1   Et   Feixd 2 

 F   E t e ixdx      2 I() I() Isin  I   F m  a  sin  a

Grating performs a Fourier transform - intensity distribution on screen is the Fourier image of the incoming time-dependent EM field Pulse shaping.

Et Ite it  ESe    i

EoutthtEt  in   Altering E(t) - will cause change in pulse shape and in spectrum  EHEout    in   Altering E() - will cause change in spectrum and in pulse shape

www.physics.gatech.edu/gcuo/lectures/UFO13Pulseshaping.ppt Pulse shaping: optical Fourier transform

S() S’() H( )   

x ()x

grating grating f f ff f f John Heritage, UC Davis Fourier Andrew Weiner, Purdue Transform Plane How it works: The grating disperses the light, mapping color onto angle. The first lens maps angle (hence wavelength) to position. The second lens and grating undo the spatio-temporal distortions. Pulse shaping: optical Fourier transform

Instead of amplitude mask - can use phase mask, or combination of the two

Spatial modulator Pulse shaping for telecommunication.

The goal: multiple pulses with variable separations

Real example