Fourier Transform with Lenses

Fourier analysis in linear systems

 

 

1

( , ) ( , )exp 2 ( )

( , )

x y x y x y

x y

f x y g f f j xf yf df df

g f f

 





   

= F

 

 

( , ) ( , )exp 2 ( )

( , )

x y x y

g f f f x y j f x f y dxdy

f x y

 

   



 

F

( , ) ( , )

( , ) ( , )

FT IFT

x y

x y

f x y g f f

f x y g f f





Introduction to Fourier Optics, J. Goodman Fundamentals of Photonics, B. Saleh &M. Teich

Properties of 1D FT

Properties of 1D FT

Some frequently used functions

Some frequently used functions

Time duration and spectral width

The power rms width

(most of the measurement quantities)

The rms width

(Principles of optics 7^thEd, 10.8.3, p615)

Time duration and spectral width

Widths at 1/e, 3-dB, half-maximum

1 f(t)

t

= 2

2D Fourier transform

Superposition of plane waves

Properties of 2D FT

Properties of 2D FT

Properties of 2D FT

Fourier and Inverse Fourier Transform



 _ 

( f_x, f_y)

Input placed against lens

Input placed in front of lens

Input placed behind lens

back focal plane

Fourier Transform with Lenses

R₁>0 (concave) R₂<0 (convex)

 ^{x, y  kn} ^{x, y  k}^₀ ^{ } ^{x, y} 



 ^{x y} ^jk  ^jkⁿ   ^{x y} 

t_l ,  exp ₀ exp 1  ,

     ^{x y t x y U x y}

U_l^' ,  _l , _l ,

  









   

 









   







 ₂

2 2 2

2 2 1

2 2

1

0 1 1 1 1

, R

y R x

R y R x

y x

A thin lens as a phase transformation

 

' ,

Ul x y

 ^, 

Ul x y

Intro. to Fourier Optics, Chapter 5, Goodman.

The Paraxial Approximation

 

 



 



 



 



 

 







2 1

2 2

0

1 1

1 2 exp

exp

, R R

y n x

jk jkn

y x t_l

  _



 



 





2 1

1 1 1

1

R n R

f

concave :

 0 f

convex :

 0 f

   



 



 

 ^{2 2}

exp 2

, x y

f j k

y x t_l

 Phase representation of a thin lens (paraxial approximation) focal length

Types of Lenses

convex :

 0 f

concave :

 0 f

 

 



 



 

 ^{2 2}

exp 2

, x y

f j k y

x t_l

Collimating property of a convex lens

Fig. 1.21, Iizuka

z_i

Plane wave!

How can a convex lens perform the FT

f_o f_o

Fourier transforming property of a convex lens

The input placed directly against the lens

Pupil function ;   1 in sid e th e len s a p ertu re , 0 o th erw ise

P x y 

 

       

' 2 2

, , , exp

l l 2

U x y U x y P x y j k x y f

 

   

 

   

    ^{ }

2 2

' 2 2

exp 2 2

, , exp exp

f l 2

j k u

f k

U u U x y j x y j xu y dxdy

j f f f

 

 

 





 

      

 

      

   

 

   

     

2 2

exp 2 2

, , , exp

f l

j k u

U u f U x y P x y j xu y dxdy

j f f

 

 

 





 

    

 

   

 

 

Quadratic phase factor From the Fresnel diffraction formula ( z = f ):

Fourier transform

U_l U_l’

Fourier transforming property of a convex lens

The input placed in front of the lens

   

   

2 2

exp 1

2 2

, , exp

f l

k d

A j u

f f

U u U x y j xu y dxdy

j f f

 

 

 





   

 

     

 

   

 

 

If d = f

 ^,  ^{, exp 2}  

f l

U u A U x y j xu y dxdy

j f f

  

 





 

   

 

 

Exact Fourier transform !

   

d f d

j d u j k A

u U _f





 ^

 

 



2 2

exp 2

,       



 





 u d d

j d d

f d

P f t_A



 



 



 ^ 





exp 2 ,

,

Fourier transforming property of a convex lens

The input placed behind the lens

Scaleable Fourier transform !

By decreasing d, the scale of the transform is made smaller.

 ^{   } ^{ }  ^{ ,}

exp 2 ,

, ^{2 2}

0 tA

d j k d

f d

P f d U Af







 

 



 



 

Invariance of the input location to FT

Imaging property of a convex lens

magnification

From an input point S to the output point P ;

Fig. 1.22, Iizuka

Diffraction-limited imaging of a convex lens

From a finite-sized square aperture of dimension a x a to near the output point P ;

FT in cylindrical (polar) coordinates

In rectangular coordinate

In cylindrical coordinate ( , )

( , ) x y r 

( , ) ( , )

x y

f f

 

FT in cylindrical coordinates

FT in cylindrical coordinates

(Ex) Circular aperture : for the special case when

Special functions in Photonics

Special functions in Photonics

Special functions in Photonics

Appendix : Linear systems

Appendix : Shift-invariant systems

Appendix : Linear shift-invariant causal systems

p.180

Example : The resonant dielectric medium

Susceptibility of a resonant medium :

Let, Response to harmonic (monochromatic) fields :

Appendix : Transfer function

Homework

Show the FT properties