interferometers and interferometry

Download Interferometers and Interferometry

Post on 06-Nov-2015

5 views

Category:

Documents

2 download

Embed Size (px)

DESCRIPTION

Advanced

TRANSCRIPT

  • Interferometers and Interferometry

    Plane wave interference.

    Spherical wave interference

    Division of Wavefront Interferometers 2-slit interference, Lloyds mirror, biprism

    N-slit interference, Array theorem

    Division of Amplitude InterferometersMach-Zehnder, Michelson, Sagnac

    beamsplitters and multiple reflection artifacts

    Shearing Interferometers. Wedged collimation tester

    Polarization Interferometry

    Temporal Coherence and Fourier Transform SpectroscopyKelvin Wagner, University of Colorado Physical Optics 2011 1

    Plane Waves

    direction cosines of a plane wave

    E(x, y, z) = E0pei2n0

    (x+y+z)= E0pe

    i(kxx+kyy+kzz)

    2 + 2 + 2 = 1 k2x + k2y + k

    2z = n

    2k20 =

    (2n

    0

    )2where k0 = |~k| = 2n/ in medium of index n.

    x

    z

    k

    k-space

    k

    2pin/

    In 2-dimensionsE(x, z; t) = A0pe

    ik0(x sin +z cos )ei2t + cc

    where = sin / and = cos /. 5 1014 Hz .63 106m (HeNe)

    Kelvin Wagner, University of Colorado Physical Optics 2011 2

    Interference

    Intensity = |Field|2

    E1(~r, t) = A1p1ei(~k1~r1t) + cc

    E2(~r, t) = A2p2ei(~k2~r2t) + cc

    k-space2pin/

    ko

    kr

    Kg2E Eo r

    E +Eo r2 2

    I(x)

    X

    I(x) = |E1(~r, t) + E2(~r, t)|2= |A1|2 + |A2|2 + A1A2(p1 p2)ei[(~k1~k2)~r+(12)t] + cc= |A1|2 + |A2|2 + A1A2(p1 p2) cos

    [(~k1 ~k2) ~r + (1 2)t

    ]Typically require and define the followingp1 p2 = 1 Co-polarized1 2 = 0 Same FrequencyA1A

    2 = a1a2e

    i(12) = a1a2ei amplitude and phase~KG = ~k1 ~k2 = 2 KG Grating wave vectorm = ImaxIminImax+Imin =

    2I1I2

    I1+I2(p1 p2) modulation depth

    I(~r) = I1 + I2 + 2I1I2 cos

    [~Kg ~r +

    ]= I0

    (1 +m cos

    [~Kg ~r +

    ])Kelvin Wagner, University of Colorado Physical Optics 2011 3

    Spherical Waves

    x

    z eikr

    x

    z e-ikr

    Isophase surfaces are spherical, (r) = const, where r2 = x2 + y2 + z2

    Nonparaxial Spherical Wave

    A(r, t) =Aoreikreit + cc =

    Aoreik

    x2+y2+z2eit + cc

    Paraxial Regimez max(x, y) so that (x2 + y2)/z2 1

    r = z

    1 +

    x2 + y2

    z2 z + x

    2 + y2

    2z

    using1 + = 1 + /2 2/8 + ...

    Paraxial Focusing Sphereical Wave

    A(r, t) = Aoei(kzt)eik

    x2+y2

    2z + cc

    Kelvin Wagner, University of Colorado Physical Optics 2011 4

  • Spherical Wave Interference

    I(~r, t) =

    p1(~r ~r1) A1|~r ~r1|ei(k|~r~r1|t) + p2(~r ~r1) A2|~r ~r2|ei(k|~r~r2|t)2

    =A21R21

    +A22R22

    + 2 (p1(R1) p2(R2))A1R1

    A2R2

    cos [k(R1 R2) ]

    Ri = |~r ~ri|k(R1 R2) = 2n maximak(R1 R2) = (2n + 1) minima

    R1 R2

    Kelvin Wagner, University of Colorado Physical Optics 2011 5

    Parabolic Approximation

    A(x, y, z, t) = eik

    x2+y2+z2eitE(R)

    when Z max(x, y) x2+y2z2

    1 can make Fresnel approxA(x, y, z, t) =

    a0zei(kzt)ei

    k2z (x

    2+y2)

    2 point sources in paraxial regime

    A1(x, y, z, t) =a1zei(kzt)ei

    k2z

    ((xd2)

    2+y2

    )

    A2(x, y, z, t) =a2zei(kzt)ei

    k2z

    ((x+d2)

    2+y2

    )

    Interference gives intensity x2dx+d24 (x2+dx+d2

    4 )

    I(x, y, z) =a1z

    2 + a2z

    2 = a1a2z2

    ei k2z

    [(xd2)

    2(x+d2)2+y2y2

    ]

    =1

    z2

    (|a1|2 + |a2|2 + 2(a1a2) cos

    (k

    2z2dx + (a1a

    2)

    ))

    d

    d

    Kelvin Wagner, University of Colorado Physical Optics 2011 6

    Youngs Double Slit

    PointSource

    uniform coneangle

    d/2-d/2

    R1

    R2

    z

    xy

    radiusa

    R1 =

    (x d

    2

    )2+ y2 + z2 R2 =

    (x +

    d

    2

    )2+ y2 + z2 R22R21 = 2xd

    R = R2 R1 = (R2 R1) (R2 +R1)R2 +R1

    =R22 R21R2 +R1

    =2xd

    R2 +R1Fringes strongest near x, y = 0 especially for sources with noticable bandwidth, largepinholes, or large sources

    R2 +R1 2z R = xdz

    I(x, y) =P

    4A2

    4a2

    (I1 + I2 + 2

    I1I2 cos

    (2

    xd

    z

    ))Separation of adjacent fringes x = zd varies with wavelength

    Kelvin Wagner, University of Colorado Physical Optics 2011 7

    Slit Width

    k = 22xz

    I(x) =

    d+wo/2dwo/2

    I0(1 + cos kx)dx = I0wo +

    (x dwo

    )cos kxdx

    = I0wo + I0wosinc kwo cos kd = I0wo

    (1 + sinc

    2

    2x

    zwo cos

    2

    2x

    zd

    )Visbility= ImaxIminImax+Imin =

    sinc 2 2xz woHigh visibility for w < z

    24x = wmax

    Kelvin Wagner, University of Colorado Physical Optics 2011 8

  • Polychromatic Illumination

    Combination of mutually incoherent monochromatic componentsEach component produces a fringe pattern of varying scale

    Sum to get total intensity

    Fringe spacing between maxima s = z/d

    For a center frequency 0and bandwidth fringes will blur out if x > s/4

    x

    x0max

    x x0 0

    t(t)

    0

    2pi

    (t)-(t-)

    0

    2pi

    2pi

    |()|

    00

    Kelvin Wagner, University of Colorado Physical Optics 2011 14

    Spectra of the Source

    E() = 12

    E(t)eitdt =1

    2E0( 0)

    ei(t)eitdt

    =1

    2E0( 0)

    n

    (n+1)0n0

    eineitdt =E02( 0)

    n

    ein0sinc0

    Power SpectraP () = |E()|2 = |E0|2 20 sinc 2[( 0)0]

    Spectral width = 2

    = 10

    Coherence time 0Coherence length c0 =

    c =

    =

    2

    = Lc

    Lc Nwaves

    5000A 1A 2mm 5000

    5500A 1250A 2m 4

    6328A 107A = 7.5KHz 40km 6.3 10106328A 6.67 105A = 500MHz .6m 106

    Kelvin Wagner, University of Colorado Physical Optics 2011 15

    Source with random phase jumps at equalintervals

    Kelvin Wagner, University of Colorado Physical Optics 2011 16

  • Source with random phase jumps at randomintervals

    Kelvin Wagner, University of Colorado Physical Optics 2011 17

    Coherence Length

    t = 1/ coherence time = 1/t source bandwithl = ct coherence length

    c =200

    For an OPD = the phase difference = k0

    Fringe pattern

    I() = 2

    i(k0)(1 + cos k0)dk0

    V = ImaxIminImaxr+Imin

    Fringe modulation depth, m (aka visibility) goes to 0 at

    x0 =2

    k=

    c

    = l = ct

    Kelvin Wagner, University of Colorado Physical Optics 2011 18

    Fourier Transform Spectroscopy

    L /2

    Mirror

    Translating Mirroron precision rail

    L /21

    2 z

    DetectorI(z)

    Fringes I( ) = 2I1 [1 + ( ) cos (arg (( )))]

    Integrate interference over all wave-numbers to give FT of band shape envelope B(k)

    I(z) =

    0

    B(k)(1 + cos kz)dk = I0 +1

    2

    B(k)eikzdk

    So by subtracting out DC term, FFT1 fringes we can get spectrum

    B(k) =

    [I(z) I0]eikzdz

    Kelvin Wagner, University of Colorado Physical Optics 2011 19

    Coherence Functions for typical laser sources

    Spectrum I() Complex Coherence () |()| Fringe Visibility V () (I1 = I2)True Monochromatic Waves

    I()0

    V()I() = I0( 0) ei2pi0

    2 Monochromatic Components

    I()1 2

    V()

    ( )2 11 2

    I() = I1( 1) + I2( 2) I1I1+I2ei2pi1 + I2I1+I2ei2pi21 4I1I2

    I1+I22 sin2[(2 1) ]

    Doppler Broadened Line

    I() = 2I0D

    ln 2

    pie

    2(0)D

    ln 2

    2

    I()0

    Area=I0ei2pi0e

    hpiD

    2ln 2

    i2e

    hpiD

    2ln 2

    i2

    V()

    where D =0c22k ln 2

    TM

    T = temperature KM = mass of one atom Kgk = Boltzman 1.38 1023 JKTwo Doppler Broadened Lines of equal intensity and equal width

    I()= I0D

    ln 2

    pi

    [e

    2(1)D

    ln 2

    2

    I()1 2

    | cos[(2 1) ]ehpiD

    2ln 2

    i2

    V()

    ( )2 11 2 30+e

    2(2)D

    ln 2

    2]

    Lorentzian line

    I()20

    Area=I0 V()

    1/pi

    1I() = I0

    2pil

    1+

    h0L

    i2 r ei2pi0epil epil

    Kelvin Wagner, University of Colorado Physical Optics 2011 20

  • Generalized Imaging ModelDiffraction and Aberration Effects

    EntrancePupilObject Image

    xy

    xy

    x0

    zo zi

    S

    S

    xs

    yt

    x

    y

    x

    y

    Exit Pupil withPhase Aberration

    y0h

    Mh

    Abbe TheoryOnly some components of object spectrum are captured by the entrance pupil. Highfrequency components are blocked giving limit s on image resolution.

    Rayleigh TheoryDiffraction effects resulting from finite exit pupil with imposed phase aberrations.

    Kelvin Wagner, University of Colorado Physical Optics 2011 21

    Aberrations

    Aberration Free SystemExit pupil illuminated by perfect sphericalwave that focusses toward the geometric imageAberrated Imaging System

    Aberrations can be modeled as a complex phas shiftingplate at the exit pupil.

    P(x, y) = P (x, y)eikW (x,y)W (x, y) is the effective path length errorCan be characterized by peak-to-valley or RMS OPDRMSOPD = 1

    A

    AW (x, y)

    2dA

    CTF in the presence of aberrations

    H(fx, fy) = P (zifx,zify)eikW (zifx,zify)

    Impulse Response

    h(x, y) =1

    (zi)2p

    (x

    zi ,y

    zi

    )F

    {eikW (zifx,zify)

    }

    Peak to Valley OPD

    Reference Sphere

    Off-axis aberrated wavefront

    Kelvin Wagner, University of Colorado Physical Optics 2011 22

    Focussing Error

    Phase distribution across exit pupil to focus to ideal image plane zi

    i(x, y) =

    zi(x2 + y2)

    Misfocussed spherical wave focussing to distance za

    A(x, y) =

    za(x2 + y2)

    Path length error

    W (x, y) = k1(A i) = k1

    (1

    za 1zi

    )(x2 + y2) = 1

    2

    (1

    za 1zi

    )(x2 + y2)

    Aperture of width 2w0, maximum error is (f0 =w0zi

Recommended

View more >