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  • 4. Fresnel and Fraunhofer diffraction4. Fresnel and Fraunhofer diffraction

  • Remind! Diffraction under paraxial approx.Remind! Diffraction under paraxial approx.

  • Every unobstructed point of a wavefront, at a given instant in time, serves as a source of secondary wavelets (with the same frequency as that of the primary wave). The amplitude of the optical field at any point beyond is the superpositionof all these wavelets (considering their amplitude and relative phase).

    Huygenss principle:By itself, it is unable to account for the details of the diffraction process.It is indeed independent of any wavelength consideration.

    Fresnels addition of the concept of interference

    Remind! Huygens-Fresnel principleRemind! Huygens-Fresnel principle

  • Fresnels shortcomings :He did not mention the existence of backward secondary wavelets,however, there also would be a reverse wave traveling back toward the source.He introduce a quantity of the obliquity factor, but he did little more than conjecture about this kind.

    Gustav Kirchhoff : Fresnel-Kirhhoff diffraction theoryA more rigorous theory based directly on the solution of the differential wave equation.He, although a contemporary of Maxwell, employed the older elastic-solid theory of light.He found K() = (1 + cos )/2. K(0) = 1 in the forward direction, K() = 0 with the back wave.

    Arnold Johannes Wilhelm Sommerfeld : Rayleigh-Sommerfeld diffraction theoryA very rigorous solution of partial differential wave equation.The first solution utilizing the electromagnetic theory of light.

    Remind! After the Huygens-Fresnel principleRemind! After the Huygens-Fresnel principle

  • ( ) ( ) ( ) dsrjkrPU

    jPU

    cosexp1

    01

    0110

    =

    ( ) ( ) ( )dsPUPPhPU 1100 ,

    =

    ( ) ( )

    cosexp1,01

    0110 r

    jkrj

    PPh =

    )90by phaseincident the(lead 1 :

    1

    o

    jphase

    amplitude

    Each secondary wavelet has a directivity pattern cos

    Remind! Huygens-Fresnel Principlerevised by 1st R S solution

    Remind! Huygens-Fresnel Principlerevised by 1st R S solution

    Impulse response function

    H-F principle

  • ( ) ( ) ( ) dsr

    jkrPUj

    PU cosexp1

    01

    0110

    =

    01

    cosrz

    =

    ( ) ( ) ( )

    ddr

    jkrUjzyxU exp,, 2

    01

    01

    =

    ( ) ( )22201 ++= yxzr

    Note! So far we have used only two assumptions (R-S solution) :

    Scalar wave equation + ( r01 >> )

    Huygens-Fresnel PrincipleHuygens-Fresnel Principle

    Screen (x,y)Aperture (,)

    r01

  • +

    +

    22

    01 21

    211

    zy

    zxzr

    ( ) ( ) ( ) ( )[ ]

    ddyxz

    kjUzj

    eyxUjkz

    2

    exp,, 22

    +=

    ( ) ( ) ( ) ( ) ( )

    ddeeUezj

    eyxUyx

    zj

    zkjyx

    zkjjkz

    +++

    =2

    222222

    ,,

    ( ) ( ) ( )

    zyfzxf

    zkjyx

    zkjjkz

    YX

    eUezj

    eyxU

    /,/

    222222

    ,),(==

    ++

    = F

    Since k is 10Since k is 1077 order, the quadratic terms in rorder, the quadratic terms in r0101 must be considered in the exponent.must be considered in the exponent.

    Fresnel diffraction integral

    Fresnel (paraxial) ApproximationFresnel (paraxial) Approximation

  • This is most general form of diffraction No restrictions on optical layout

    near-field diffraction curved wavefront

    Analysis somewhat difficult

    Fresnel DiffractionFresnel Diffraction

    Screen

    zz

    Curved wavefront(diverging wavelets)

    Fresnel (near-field) diffractionFresnel (near-field) diffraction

    ( ) ( )2 2

    2( , ) ,kjzU x y U e

    +

    F

  • y

    Wavefront emitted earlier

    z Wavefront emitted

    later

    z

    y k

    Wavefront emitted

    later

    Wavefront emitted earlier

    ( )01exp jkr( )

    + 22

    2exp yx

    zkj

    ( )01exp jkr

    ( )

    + 22

    2exp yx

    zkj

    ( )yj 2exp

    ( )yj 2exp

    z=0

    z=0

    diverging

    converging

    Positive phase and Negative phasePositive phase and Negative phase

  • 16

    2Dz

    ( ) ( )[ ]2max223 4

    + yxz

    Accuracy can be expected for much shorter distances

    ( , ) 2 4for U smooth & slow varing function; x D z =

    Fresnel approximation

    Accuracy of the Fresnel ApproximationAccuracy of the Fresnel Approximation

  • ( ) ( ) ( ) ddyxhUyxU , ,, =

    ( ) ( )

    += 222

    exp, yxz

    jkzj

    eyxhjkz

    This convolution relation means that Fresnel diffraction is linear shift-invariant.

    ( ) ( ) ( ) ( )[ ]

    ddyxz

    kjUzj

    eyxUjkz

    2

    exp,, 22

    +=

    Impulse Response and Transfer Functions of Fresnel diffraction

    Impulse Response and Transfer Functions of Fresnel diffraction

    Impulse response function of Fresnel diffraction

  • ( ) ( ) ( )2 2, exp 2 1X Y X YzH f f j f f

    =

    ( ) { }

    ( ) ( ),

    2 2 2 2

    ( , )

    exp exp

    X Y

    jkzjkz

    X Y

    H f f h x y

    e j x y e j z f fj z z

    = + = +

    F

    F

    ( ) ( ) ( ) ( )2 2

    2 21 12 2

    X YX Y

    f ff f

    Under the paraxial approximation, the transfer function in free space becomes

    Impulse Response and Transfer Functions of Fresnel diffraction

    Impulse Response and Transfer Functions of Fresnel diffraction

    ( ) ( ) ( ) ( )2 2, ; , ;0 exp 2 1X Y X Y X Yzf f z f f j f f

    =

    Remind! Transfer function of wave propagation phenomenon in free space

    ( ) ( )2

    2 2, exp

    j z

    X Y X YH f f e j z f f = +

    We confirm again that the Fresnel diffraction is paraxial approximation

  • zy

    zxzr 01

    ( ) ( ) ( )

    ddeUzj

    eyxUyx

    zjjkz +

    =

    2

    ,,

    ( ){ }zyfzxf

    jkz

    YXU

    zje

    /,/,

    === F

    ( ) ( )22201 ++= yxzr

    Fresnel Diffraction between Confocal Spherical SurfacesFresnel Diffraction between Confocal Spherical Surfaces

    The Fresnel diffraction makes the exact Fourier transforms between the two spherical caps.

  • In summary, Fresnel diffraction is

  • ( ) ( ) ( )

    ddyx

    zjU

    zjeeyxU

    yxz

    kjjkz

    +=

    +2exp,,

    )(2

    22

    ( )2

    max22 +

    kz

    ( ){ }zyfzxf

    yxz

    kjjkz

    YXU

    zjee

    /,/

    )(2

    ,

    22

    ==

    +

    = F

    formular sdesigner' anttena : Dz

    22>

    Fraunhofer ApproximationFraunhofer Approximation

    01 1x yr zz z

  • Specific sort of diffraction far-field diffraction plane wavefront Simpler maths

    Fraunhofer (far-field) diffractionFraunhofer (far-field) diffraction

    ( ) ( ) ( )2, , expU x y U j x y d dz

    +

    Plane waves

  • Circular aperture 1 1

    r arcircr aa = >

  • Circular Aperture

    Circular aperture

    Single slit(sinc ftn)

    Sin 0 /D 2/D 3/D/D2/D3/D

    Intensity

  • Airy Disc Similar pattern to single slit

    circularly symmetrical First zero point when kD sin = 3.83

    sin = 1.22/D Central spot called Airy Disc

    Every optical instrument microscope, telescope, etc.

    Each point on object Produces Airy disc in image.

  • (Appendix : Step and repeat function)

  • Two-dimensional array of rectangular apertures

  • Two-dimensional array with irregular spacing

    randomized

  • Diffraction from an array of finite size

  • Diffraction from an array of finite size (2) : for large C

    unit cell (a x l)

    c

  • Double-slit diffraction

    ( ) ( ) ( ) ( )[ ]

    [ ]

    22

    222

    2

    21

    21

    2/sin)(2/sin)(2/sin)(2/sin)(

    2)(

    2)(sin2)(

    2)(sin

    cossin4cossin2

    2

    2

    cossin2

    )()(2

    sin,sin

    sin1

    =

    =

    =

    =+=

    +=

    +

    =

    ++

    +

    +

    oo

    Lo

    Ro

    o

    Liiiiii

    o

    LR

    baikbaikbaikbaik

    o

    L

    ba

    baiks

    o

    Lba

    baiks

    o

    LR

    Ir

    bEc

    irradiance ; Ec

    I

    rbE

    eeeeeeib

    rE

    E

    ka kb

    eeeeikr

    E

    dserE

    dserE

    E

    a

    b

    b

    (a-b)/2(a+b)/2

  • Double-slit diffraction

    bmpa :(2) & (1) satisfying or orders,missing for Condition

    2, 1, 0,m ,bm :minima ndiffractio

    (2) -------- - b

    b

    : termenvelope ndiffractio The

    2, 1, 0,p ,ap :maxima ce interferen

    (1) --------- a : termce interferen The

    rbEc

    I IIo

    Looo

    =

    ==

    ==

    =

    =

    =

    sin

    sin

    sinsin

    sin

    sincoscos

    2,cossin4

    2

    22

    22

    2

  • Multiple-slit diffraction

    [ ][ ]

    [ ][ ]{ }

    22

    2/

    1

    2))12(

    2))12(

    2))12(

    2))12(sinsin

    sinsinsin

    =

    += =

    +

    +

    NII

    dsedserE

    E

    o

    N

    j

    baj

    baj

    baj

    bajiksiks

    o

    LR

    integers other all p for occurminima 2N, N, 0,p for occurmaxima principal

    2, 1, 0,p ,aNp or ,

    Np :minima

    NNNN :magnitude

    2, 1, 0,m ,am or ,m :maxima principal

    (1) --------- N : ter

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