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  • 13. Fresnel diffraction

  • Remind! Diffraction regimes

  • ( ) ( )0 2exp

    , ikrzEE x y d d

    i r

    =

    ( ) ( )2 22r z x y = + + Screen (x,y)Aperture (,)

    r

    ( ) ( )00exp

    ( )ikrEE P F dA

    i r

    =

    Fresnel-Kirchhoff diffraction formula

    ( ) cos zFr

    = =

    ( ) ( )2 22 2

    2 2 2 2

    1 112 2 2 2

    2 2 2 2

    x yx yz zz z z z

    x y x yzz z z z z z

    + + = + +

    = + + + + +

    Obliquity factor :

    ( ) ( ) ( )

    ( ) ( )

    2 20

    2 2

    , exp exp2

    exp exp 2

    E kE x y ikz i x yi z z

    k ki i x y d dz z

    = + + +

  • Screen (x,y)Aperture (,)

    r( ) ( ) ( )

    ( ) ( )

    ( ) ( )

    2 20

    2 2

    2 2

    , exp exp2

    exp exp 2

    exp exp 2

    E kE x y ikz i x yi z z

    k ki i x y d dz z

    k kC i i x y d dz z

    = + + +

    = + +

    ( ) ( ) ( )2 2, ( , ) exp exp 2k kE x y C U i i x y d dz z

    = + +

    ( ) ( )2 2

    2( , ) ,kjzE x y U e

    +

    F

    ( ) ( )

    ( )

    , ( , ) exp

    ( , ) exp sin sin

    kE x y C U i x y d dz

    C U ik d d

    = +

    = +

    ( ){ }( , ) ,E x y U F

    Fresnel diffraction

    Fraunhofer diffraction

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

    near-field diffraction curved wavefront

    Analysis somewhat difficult

    Fresnel Diffraction

    Screen

    z

    Curved wavefront(parabolic wavelets)

    Fresnel (near-field) diffraction

    ( ) ( )2 2

    2( , ) ,kjzU x y U e

    +

    F

  • 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 Approximation

  • In summary, Fresnel diffraction is

  • 13-7. Fresnel Diffraction by Square Aperture

    (b) Diffraction pattern at four axial positions marked by the arrows in (a) and corresponding to the Fresnel numbers NF=10, 1, 0.5, and 0.1. The shaded area represents the geometrical shadow of the slit. The dashed lines at represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide with the edges of the geometrical shadow, the Fresnel number NF=0.5.

    ( )dDx /=

    Fresnel Diffraction from a slit of width D = 2a. (a) Shaded area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffracted beam.

    2 / : Fresnel numberFN a z=

    2w2a

  • ( ) ( ) ( ) ( )[ ]

    ddyxz

    kjUzj

    eyxUjkz

    2

    exp,, 22

    +=

  • 2 / : Fresnel numberFv N a z = =

  • Fresnel diffraction from a wire

  • Fresnel diffraction from a straight edge

  • From Huygens principle to Fresnel-Kirchhoff diffraction

  • Huygens principle

    Given wave-front at t

    Allow wavelets to evolve for time t

    r = c t

    New wavefront

    What about r direction?(-phase delay when the secondary wavelets, Hecht, 3.5.2, 3nd Ed)

    Construct the wave front tangent to the wavelets

    Every point on a wave front is a source of secondary wavelets.i.e. particles in a medium excited by electric field (E) re-radiate in all directionsi.e. in vacuum, E, B fields associated with wave act as sources of additional fields

    secondary wavelets

    Secondarywavelet

  • Huygens wave front construction

  • Incompleteness of Huygens principle

    Fresnels modification Huygens-Fresnel principle

  • Huygens Secondary wavelets on the wavefront surface O

    O

    P

    Spherical wave from the point source SObliquity factor:

    unity where =0 zero where = /2

    Huygens-Fresnel principle

    daFer

    er

    EdaFerr

    EE ikrA

    ikrs

    rrik

    Asp

    pp

    )(1'

    1)('

    1 ')'(

    == +

  • Kirchhoff modificationFresnels 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-Kirchhoff 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 F() = (1 + cos )/2. F(0) = 1 in the forward direction, F() = 0 with the back wave.

  • ( )

  • /2

    Z1

    Z2 Z3

    Spherical wave from source Po

    Huygens Secondary wavelets on the wavefront surface S

    Obliquity factor: unity where =0 at C zero where =/2 at high enough zone index

    : Fresnel Zones

  • : Fresnel Zones

    The average distance of successive zones from P differs by /2 -> half-period zones.Thus, the contributions of the zones to the disturbance at P alternate in sign,

    Z1

    Z2 Z3

    For an unobstructed wave, the last term n=0.

    Whereas, a freely propagating spherical wave from the source Po to P is

    Therefore, one can assume that the complex amplitude of

    1 exp( )iksi s

    =

    (1/2 means averaging of the possible values,more details are in 10-3, Optics, Hecht, 2nd Ed)

  • : Diffraction of light from circular apertures and disks

    (a) The first two zones are uncovered,

    (b) The first zone is uncovered if point P is placed father away,

    (c) Only the first zone is covered by an opaque disk,

    12

    1

    1

    : Babinet principle

    RVariation of on-axis irradiance Diffraction patterns fromcircular apertures

    RP

    (consider the point P at the on-axis P)

    112

  • Fresnel diffraction from a circular aperture

    Poisson spot

  • Babinet principle

    At screen At complementaryscreen

    { }Amplitude of S

    { }Phase of S

    { }Amplitude of CS S CS UN + =

    { }Phase of CS

    S CS

    without screen

  • : Straight edge

    Damped oscillatingAt the edge

    Monotonically decreasing

  • 13-6. The Fresnel zone plateThe average distance of successive zones from P differs by /2 -> half-period zones.Thus, the contributions of the zones to the disturbance at P alternate in sign,

    Assumeplane wavefronts

    22 22 2 2

    0 0 00 02 4

    nnR r n r r n

    r r = + = +

    ( )0 0 nR nr r >>

    R1O

    R3

    R4

    RN

    R2

    P

    0 2r N +

    0r

    Z1

    Z2 Z3

    If the even zones(n=even) are blocked 1 3 5( )P = + + + Bright spot at P

    It acts as a lens!

  • 2

    0nRr

    n=

    ( )0 0 nR nr r >>

    R1O

    R3

    R4

    RN

    R2

    P

    0 2r N +

    0r2

    11 0 ( 1)

    Rf r n

    = = =

    Fresnel zone-plate lens

    0 11

    2 2nnR r Rn n

    = =nR

    ( )sin sin tan nn m m mm n

    R mR mf R

    = = =

    ( )( ) ( ) 111 12m n nRf R R nR

    m mn = =

    21

    mRfm

    =

    1f2f3f

    Fresnel zone-plate lens has multiple foci.

  • Sinusoidal zone plate: This type has a single focal point.

    Binary zone plate: The areas of each ring, both light and dark, are equal.It has multiple focal points.

    Fresnel zone-plate lens

    For soft X-ray focusing

    Fresnel lens: This type has a single focal point.Focusing efficiency approaches 100%.

    http://en.wikipedia.org/wiki/Image:Zonenplatte_Cosinus.pnghttp://en.wikipedia.org/wiki/Image:Zone_plate.svghttp://en.wikipedia.org/wiki/Image:Fresnel_lens.svg

    1 2 3 4 5 6 713-7. Fresnel Diffraction by Square Aperture 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31