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Transcript of Fourier transformFourier transform - shsong/3-Fourier to Fourier Optics, J

• Fourier transformFourier transform

{ }{ }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 yg 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. GoodmanFundamentals of Photonics, B. Saleh &M. Teich

• Properties of 1D FTProperties of 1D FT

• Properties of 1D FTProperties of 1D FT

• Some frequently used functionsSome frequently used functions

• Some frequently used functionsSome frequently used functions

• Time duration and spectral widthTime duration and spectral width

The power rms width(most of the measurement quantities)

The rms width

(Principles of optics 7th Ed, 10.8.3, p615)

• Time duration and spectral widthTime duration and spectral width

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

1

f(t)

t

= 2.

• 2D Fourier transform2D Fourier transform

Superposition of plane waves

• Remind ! Spatial frequency and propagation angleRemind ! Spatial frequency and propagation angle

z

directional cosine : x =

1

x =

• Spatial frequency and propagation angleSpatial frequency and propagation angle

• Fourier and Inverse Fourier Transform

( , )x yf f

• Properties of 2D FTProperties of 2D FT

• Properties of 2D FTProperties of 2D FT

• FT in cylindrical (polar) coordinatesFT in cylindrical (polar) coordinates

In rectangular coordinate

In cylindrical coordinate

( , )( , )x yr

( , )

( , )x yf f

• FT in cylindrical coordinatesFT in cylindrical coordinates

• FT in cylindrical coordinatesFT in cylindrical coordinates

(Ex) Circular aperture : for the special case when

• Special functions in PhotonicsSpecial functions in Photonics

• Special functions in PhotonicsSpecial functions in Photonics

• Special functions in PhotonicsSpecial functions in Photonics

• Input placed

against lens

Input placed

in front of lens

Input placed

behind lens

back focal plane

Fourier Transform with LensesFourier Transform with Lenses

• R1>0 (concave)R2

• The Paraxial Approximation

( ) [ ] ( )

+=

21

22

011

21expexp,

RRyxnjkjknyxtl

( )

21

1111RR

nf

concave:0f

( ) ( )

+= 22

2exp, yx

fkjyxtl

Phase representation of a thin lens (paraxial approximation)

focal length

• Types of Lensesconvex:0>f

concave:0

• Collimating property of a convex lensCollimating property of a convex lens

Fig. 1.21, Iizuka

zi

Plane wave!

• How can a convex lens perform the FTHow can a convex lens perform the FT

fo fo

• Fourier transforming property of a convex lensFourier transforming property of a convex lensThe input placed directly against the lens

Pupil function ; ( ) 1 in side the lens aperture,0 otherw ise

P x y

=

( ) ( ) ( ) ( )' 2 2, , , exp 2l lkU x y U x y P x y j x yf

= +

( )( )

( ) ( ) ( )2 2

' 2 2

exp2 2, , exp exp

2f l

kj uf kU u U x y j x y j xu y dxdyj f f f

+ = + +

( )( )

( ) ( ) ( )2 2exp

2 2, , , expf l

kj uf

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

+ = +

From the Fresnel diffraction formula ( z = f ):

Fourier transform

• Fourier transforming property of a convex lensFourier transforming property of a convex lensThe input placed in front of the lens

( )( )

( ) ( )2 2exp 1

2 2, , expf l

k dA j uf f

U u U x y j xu y dxdyj f f

+ = +

If d = f

( ) ( ) ( )2, , expf lAU u U x y j xu y dxdy

j f f

= +

Exact Fourier transform !

• ( )( )

df

dj

udkjA

uU f

+

=

22

2exp

, ( ) ( ) ddud

jdf

dfPtA

+

2exp,,

Fourier transforming property of a convex lensFourier transforming property of a convex lensThe input placed behind the lens

Scaleable Fourier transform !

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

( ) ( ) ( ) , 2

exp,, 220 Atdkj

df

dfP

dAfU

+

=

• Invariance of the input location to FTInvariance of the input location to FT

• Imaging property of a convex lensImaging 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 lensDiffraction-limited imaging of a convex lens

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

• Appendix : Linear systemsAppendix : Linear systems

• Appendix : Shift-invariant systemsAppendix : Shift-invariant systems

• Appendix : Linear shift-invariant causal systemsAppendix : Linear shift-invariant causal systems

• p.180Example : The resonant dielectric medium

Susceptibility of a resonant medium :

Let,

Response to harmonic (monochromatic) fields :

• Appendix : Transfer functionAppendix : Transfer function