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

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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

    + = +

    Quadratic phase factor

    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