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

    is cutoff for square pupil)

    Wm =12

    (1

    za 1zi

    )w20 W (x, y) =

    Wmw20

    (x2 + y2)

    OTF

    GM(fx, fy) =

    A(fx,fy)eikWm

    w20

    [(x+

    zi2 fx

    )2+(y+

    zi2 fy

    )2(xzi2 fx)2(yzi2 fy)2]dx dy

    A(0,0) dx dy

    Kelvin Wagner, University of Colorado Physical Optics 2011 23

    OTF with Focussing Error: Notice phaseshift of spokes

    Kelvin Wagner, University of Colorado Physical Optics 2011 24

  • OTF with Focussing Error: 1.25 and 1.5phase of TFFS as edge of nonzero MTF

    =*

    =*

    Kelvin Wagner, University of Colorado Physical Optics 2011 25

    Geometrical Calculation of Wave Aberrations

    Consider a ref sphere of radius r centered at Po = (xo, yo, zo) that passes through x, y, zorigin in Exit pupil, and a point P = (x, y, z) on the distorted wavefront.wavefront error

    w = r PPo =x2o + y

    2o + z

    2o

    (xo x)2 + (yo y)2 + (zo z)2

    = zo

    1 +

    x2o + y2o

    z2o ((zo z)

    1 +

    (xo x)2 + (yo y)2(zo z)2

    = zo

    [SS1 +

    x2o + y2o

    2z2o

    x4o + 2x2oy2o +

    y4o8z4o

    ] zo

    [SS1 +

    x2o 2xox + x2 + y2o 2yoy + y22z2o

    18z4o

    {x4o+4x

    2x2o+x4+

    y4o+4y

    2y2o+y4+2(x2ox

    2+

    x2oy2o+x

    2oy

    2+x2y2o+x2y2+y2oy

    2)

    + 2(4xxoyyo2xx2o2x3xo2xxoy2o 2xxoy22yyox2o2yyox22yy3o2y3yo)}]

    =2xxox2+2yyoy2

    2zo+

    1

    8z4o

    [x4+y4+2x2y2 +4x3xo+4y3yo+4x2yyo+4xxoy2

    spherical coma6x2x2o6y2y2o2x2y2o 2y2x2o8xxoyyo+4xx3o+4yy3o+4yy3+4xxoy2o+4yyox2o ]

    distortion

    Kelvin Wagner, University of Colorado Physical Optics 2011 26

    Rotationally Symmetric System

    The monochrmoatic wave aberrations can be expressed as a Taylor expansion that onlydepends on the vector to the object point ~r = (x, y) and the vector ~ = (x, y) in thepupil plane at which the ray strikes. We can expand the wave aberration function as

    W (xo, yo, x, y) =k,l,m,n

    Wklmnxkylxmyn

    But when the system is rotationally symmetric, simultaneous rotation around the opti-cal axis leaves the wave aberration function W unchanged, and traditionally we rotateto bring the object to be aligned with the y axis (so that ~r = (0, h) for an off axis objectheight h, and ~ = ( sin, cos) ) making it easy to identify y z as the Meridonalplane and x z as the sagittal plane This tells us the wave aberration can only dependon the combination of coordinates invariant to rotation:~r ~r = h2, ~ ~ = 2, and ~ ~r = h cos

    piston defocus lateral mag. 3rd ord piston spherical

    W = W200h

    2 + W020

    2 + W111h cos +

    W400h

    4 + W040

    4

    +W131h3 cos + W222h22 cos2 + W220h22 + W311h3 cos

    coma astigmatism field curvature distortion

    Kelvin Wagner, University of Colorado Physical Optics 2011 27

    Converging Spherical Wave Illuminating anAperture: Scaled FT at focus

    Amplitude just after aperture of transmittance t0(x, y)

    u(x, y;d+) = Adeikdei

    k2d(x

    2+y2)t0(x, y)

    Propagate through a distance z is given by a convolution

    hz(x, y) =eikz

    izei

    k2z (x

    2+y2)

    t (x,y)0d

    x

    z0

    u(x, y; 0) = u(x, y;d+) hd(x, y)=

    eikd

    id

    u(x, y;d+)ei k2d[(xx)2+(yy)2]dxdy

    =eikd

    idei

    k2d(x

    2+y2)

    u(x, y;d+)ei k2d(x2+y2)ei2d(xx+yy)dxdy

    Fxy{u(x, y;d+)ei k2d(x2+y2)

    } u=x/dv=y/d

    =A

    id2ei

    k2d(x

    2+y2)T0

    ( xd,y

    d

    )Since quadratic phase factors cancel

    I(x, y; 0) =

    (A

    d2

    )2 T0 ( xd,y

    d

    )2Kelvin Wagner, University of Colorado Physical Optics 2011 28

  • Aberrations viewed as wavefront error andvisualized with FT that gives impulse

    response

    Aberrations describe the imperfections of lensesAlos intrinsic aberrations for off-axis free-space focusing

    Can be affected by lens choice, orientation, and alignment Can look at focus or instead viualize with interferometric wavefront Difference between actual and ideal wavefront yields interferogramMany types of interferometers can be used for such optical testing Use ideal component as reference to test unknown component

    or can use CGH reference to test singular optical component

    Can test individual lenses or entire optical system Simplest test is to use lens as a collimator

    compare collimated beam with planar wavefront

    We will explore a library of the primary Seidel aberrationsKelvin Wagner, University of Colorado Physical Optics 2011 29

    Diffraction LimitedPeak: jinc(0)2 = .616

    Kelvin Wagner, University of Colorado Physical Optics 2011 30

    Spherical Aberration: .25 waveStrehl Ratio .78

    Kelvin Wagner, University of Colorado Physical Optics 2011 31

    Spherical Aberration: .5 waveStrehl Ratio .4

    Kelvin Wagner, University of Colorado Physical Optics 2011 32

  • Spherical Aberration: 1 waveStrehl Ratio .09

    Kelvin Wagner, University of Colorado Physical Optics 2011 33

    Spherical Aberration: 2 waveStrehl Ratio .05

    Kelvin Wagner, University of Colorado Physical Optics 2011 34

    Spherical Aberration: 4 waveStrehl Ratio .026

    Kelvin Wagner, University of Colorado Physical Optics 2011 35

    MisFocus: .25 waveStrehl Ratio .95

    Kelvin Wagner, University of Colorado Physical Optics 2011 36

  • MisFocus: .5 waveStrehl Ratio .39

    Kelvin Wagner, University of Colorado Physical Optics 2011 37

    MisFocus: 1 waveStrehl Ratio .05

    Kelvin Wagner, University of Colorado Physical Optics 2011 38

    -1 wave MisFocus partially compensates1 wave Spherical : Strehl Ratio .94

    Kelvin Wagner, University of Colorado Physical Optics 2011 39

    1 wave MisFocus partially compensates 1wave

    Spherical -2 waves 5th order spherical : SR .31

    Kelvin Wagner, University of Colorado Physical Optics 2011 40

  • Spherical Aberration OTF plots fromliterature

    Optics, Born and WolfIntroduction to the Optical Transfer Function, C.S. Williams and O.A. Becklund, SPIE 1989

    Kelvin Wagner, University of Colorado Physical Optics 2011 41

    Field Curvature

    h toto ti

    ti

    h

    Curved Focal Surface

    Petzval

    totito

    ti

    Field curvature even for thin lens. Sag: t

    21

    2

    2fRadius of curvature (ht1) =

    h2

    2 f

    For j surfaces with flat object 0 =, Image curvature given by Petzval sum1j+1

    = nj+1j

    k=0

    (nk+1 nk)ck+1nk+1nk

    Kelvin Wagner, University of Colorado Physical Optics 2011 42

    Curvature of Field: 1/4 waveDiffraction limited on-axis, misfocused off-axis

    Kelvin Wagner, University of Colorado Physical Optics 2011 43

    Curvature of Field: 1/2 waveDiffraction limited on-axis, misfocused off-axis

    Kelvin Wagner, University of Colorado Physical Optics 2011 44

  • Curvature of Field: 1 wave2 wave along diagonal

    Kelvin Wagner, University of Colorado Physical Optics 2011 45

    Coma: .25 wave

    Kelvin Wagner, University of Colorado Physical Optics 2011 46

    Coma: .5 wave

    Kelvin Wagner, University of Colorado Physical Optics 2011 47

    Coma: 1 wave

    Kelvin Wagner, University of Colorado Physical Optics 2011 48

  • Coma: 2 wave

    Kelvin Wagner, University of Colorado Physical Optics 2011 49

    Coma: 4 wave

    Kelvin Wagner, University of Colorado Physical Optics 2011 50

    Coma OTF plots from literatureOptics, Born and WolfIntroduction to the Optical Transfer Function, C.S. Williams and O.A. Becklund, SPIE 1989

    Kelvin Wagner, University of Colorado Physical Optics 2011 51

    Astigmatism from Cylindrical lens or laserdiode: .5 wave

    Kelvin Wagner, University of Colorado Physical Optics 2011 52

  • 1/2 Wave Cylindrical Astigmatismcompensated by 1/2 wave misfocus

    Kelvin Wagner, University of Colorado Physical Optics 2011 53

    1/2 Wave Cylindrical Astigmatismcompensated by -1/2 wave misfocus

    Kelvin Wagner, University of Colorado Physical Optics 2011 54

    What is Astigmatism?

    Kelvin Wagner, University of Colorado Physical Optics 2011 55

    Geometry of Astigmatism

    TangentialFocus

    SagitalFocus

    Circle of leastConfusion

    Tangential Fan

    Sagital Fan

    PetzvalST

    ParaxialFocal Plane

    object

    Off-axis rays launched in tangential fan (in plane off off-axis point) or sagital fan (perpendicular plane) come to focus in different curvedfocal surfaces.

    Tangential focus where tangential features have highest resolution.

    Sagital focus where radial features have highest resolution.

    All focal surfaces converge on axis to the paraxial focus.

    Kelvin Wagner, University of Colorado Physical Optics 2011 56

  • Astigmatism and Moving the Focal Plane

    Astigmatism: Varying Focal Planes

    At Paraxial focus, any orientation is in focus, but as we move off-axis lines misfocus Sagital and Tangential features go out of focus at different rates, first T then S

    Misfocused (d = 1.5mm) brings sagital (radial) features into focus at field edge on axis not too blurry, but tangential features still blurry at field edge

    Misfocused by 3 (d = 4.5mm) brings tangential features into focus at field edge on axis totally out of focus and sagtital (eg radial) features blurry

    Kelvin Wagner, University of Colorado Physical Optics 2011 57

    Curved Focal planes, Astigmatism andPetzval surface: Placement of flat CCD

    Focal length and field curvature can vary with wavelength

    Tangential and Sagital focal surfaces (T P ) = 3(S P ) (T S) = 2(S P )

    Uncompensated No Astigmatism No Field Curvature Negative Petzval for flat averageT + S

    With balanced 4th order Field Cur-vature

    Kelvin Wagner, University of Colorado Physical Optics 2011 58

    3rd order Astigmatism : .25 wave

    Kelvin Wagner, University of Colorado Physical Optics 2011 59

    3rd order Astigmatism : .5 wave

    Kelvin Wagner, University of Colorado Physical Optics 2011 60

  • 3rd order Astigmatism : 1 wave

    Kelvin Wagner, University of Colorado Physical Optics 2011 61

    1/2 wave 3rd order Astigmatism with 1/4wave misfocus

    Kelvin Wagner, University of Colorado Physical Optics 2011 62

    1/2 wave 3rd order Astigmatism with 1/2wave misfocus

    Kelvin Wagner, University of Colorado Physical Optics 2011 63

    1/2 wave 3rd order Astigmatism with 1 wavemisfocus

    Kelvin Wagner, University of Colorado Physical Optics 2011 64

  • 1/2 wave 3rd order Astigmatism with 1/4wave curvature of field

    Kelvin Wagner, University of Colorado Physical Optics 2011 65

    1/2 wave 3rd order Astigmatism with 1/2wave curvature of field

    Kelvin Wagner, University of Colorado Physical Optics 2011 66

    Interferometric visualization of aberrations

    Kelvin Wagner, University of Colorado Physical Optics 2011 67

    Interferometric visualization of aberrations

    Kelvin Wagner, University of Colorado Physical Optics 2011 68

  • Varieties of Interferometers

    Sagnac -- rotational sensing

    BS

    null port

    Mirror

    Mirror

    Twyman-Green

    objectundertest

    MichelsonMirrorCompensatingPlate

    BS MirrorSource

    Mach-Zehnder

    BS

    BS

    objectundertest

    Kelvin Wagner, University of Colorado Physical Optics 2011 69

    Fizeau Interferometer

    Kelvin Wagner, University of Colorado Physical Optics 2011 70

    Beamsplitters

    1 t

    r

    tt t rr

    rt

    r

    tr

    HL

    L

    LH

    H

    H

    g

    h

    l

    Conservation of Energy

    |r|2 + |t|2 = 1 |r|2 + |t|2 = 1Reciprocity

    tt + rr = 1

    tr + rt = 0

    r t = /2Z.Y. Ou and L. Mandel, Am J. Phys 57(1), Jan 1989

    Kelvin Wagner, University of Colorado Physical Optics 2011 71

    Multiple Reflection Artifacts in Beamsplitters

    0

    1

    22

    1

    3

    24

    44

    3533

    Cube Beamsplitter Plate Beamsplitter

    Wedge Beamsplitter

    0

    1

    22

    13

    Rotated Beamsplitter(constant deviation reflection artifacts)

    Kelvin Wagner, University of Colorado Physical Optics 2011 72

  • Multiple Reflection Artifacts in Beamsplitters

    Michelson Interferometer MachZehnder Interferometer

    Kelvin Wagner, University of Colorado Physical Optics 2011 73

    Stokes Principle of Reciprocity

    1 r12

    t12

    1

    2 2n

    1n

    r21t12*

    *r12*r12

    *r12t12*

    t12* t21 r12

    t12

    Time reversing all the inputs and combiningshould recombine light to send it back whereit came from.

    t12t21 + r12r

    12 = 1 t

    12r21 + r

    12t

    12 = 0

    r12 = r21conservation of Energy

    r212 + t212

    n2 cos 2n1 cos 1

    = 1 = R + TFrom Fresnel relationsts12 =

    2n1 cos in1 cos i+n2 cos t

    ts21 =2n2 cos t

    n2 cos t+n1 cos i

    tp12 ==2n1 cos i

    n2 cos i+n1 cos ttp21 ==

    2n2 cos tn1 cos t+n2 cos i

    t12n1 cos 1

    =t21

    n2 cos 2t12t21 = t

    212

    n2 cos 2n1 cos 1

    = T

    Kelvin Wagner, University of Colorado Physical Optics 2011 74

    Shearing Interferometry

    Self referencing Can be used with collimated or diverging light Laser or white light Linear Shear or rotational shear

    Used to Measure

    Wavefront aberrations Lens performance and OTF Fluid and plasma diagnostics AO atmospheric compensation

    Linear Shear of Wavefront

    = (x x0, y) (x, y) ddxx0

    Interferogram

    I(x, y) = 2(1 + cos)

    Shear introduced by:

    Birefringent device Wollaston or walkoff

    Parallel Plate or wedge Mach-Zehnder Grating

    Kelvin Wagner, University of Colorado Physical Optics 2011 75

    Defocus measurement using shear plate

    Focusing or Defocusing wave has quadratic phase factor

    (x, y) = k OPD = 2z

    (x2 + y2)

    shearing gives

    (x, y) =2

    z2xx0

    Corresponds to tilted plane wave. Gives linear phase interferogram with spatial fre-quency proportional to curvature.

    I(x, y) = 1 + cos

    (2

    z2xx0

    )

    Gaussian beam: R1(z) = 2zz2+z20

    z0 =w20

    The New Physical Optics Notebook: Tutorials in Fourier Optics by G.O. Reynolds, J.B. Develis, G.B. Parrent, B. Thompson, SPIE 1989

    Kelvin Wagner, University of Colorado Physical Optics 2011 76

  • Wedged Shear Plate Collimation Tester

    Tilted Wedge

    Wedged Shear-Plate Collimation TesterCollimated

    UncollimatedWedge perpendicular

    to tilt direction

    Perpendicular to fiducial line

    Kelvin Wagner, University of Colorado Physical Optics 2011 77

    Wedged Shear Plate Aberration Testing

    Engineering Synthesis Design, Inc. 2003

    Kelvin Wagner, University of Colorado Physical Optics 2011 78

    Real-time Atmospheric sensing andcompensation

    F

    Crossed dual frequency

    shearing grating

    The New Physical Optics Notebook: Tutorials in Fourier Optics by G.O. Reynolds, J.B. Develis, G.B. Parrent, B. Thompson, SPIE 1989

    Kelvin Wagner, University of Colorado Physical Optics 2011 79

    Shack-Hartman Wavefront sensing

    Ideal PlanarWavefront for

    Callibration

    AberratedWavefront

    Lensletarray CCD

    Focal Plane

    LocalGradients

    Shack-Hartman Wavefront Sensor

    Lenslets focus segment of wavefrontWavefront tilts estimated by spot shifts Tradeoff between number of lenslets andmaximum tilt which can be measured

    Works best once well collimated Can accurately measure aberrations

    Kelvin Wagner, University of Colorado Physical Optics 2011 80

  • Shack-Hartman Wavefront sensing data in 2D

    Kelvin Wagner, University of Colorado Physical Optics 2011 81

    Zernike polynomials to represent wavefronts

    Orthogonal Polynomials in Circle. Polynomials in u, v written in , RMS wavefront error is square root of weighted sum of zernike coefficients squared

    Kelvin Wagner, University of Colorado Physical Optics 2011 82

    Fitting data to Zernike polynomials

    Aquire SH image Determine centroids Find Gradients

    Integrate gradients to get sampled representation of wavefront MSE fit to sum of Zernike polynomials to get coefficients

    Kelvin Wagner, University of Colorado Physical Optics 2011 83

    Polarization splitting Mach-ZehnderInterferometer

    PBS

    BS

    Polarization Splitting Mach-Zehnder Interferometer

    Can orthogonal polarizations interfere? How can they be made to interfere? Note that photons DO traverse both paths

    but they do it in a way that is labeled with which path encoding.

    A quantum eraser that destroys the which path encoding is needed for interferenceKelvin Wagner, University of Colorado Physical Optics 2011 84

  • Vectorial Interferences-polarized

    ks

    I(x, y) =yE1ei(kxx+kzzt) + yE2ei(kxx+kzzt)2

    modulation depth m = 2I1I2

    I1+I2

    Kelvin Wagner, University of Colorado Physical Optics 2011 85

    Vectorial Interferencep-polarized

    kp

    = kxk0 = sin =kzk0= cos

    I(x, y) =E1(x z )eik0(x+z)eit + E2(x + z )eik0(x+z)eit2

    p1 p2p1 k1 = 0 p2 k2 = 0

    modulation depth

    m =2I1I2

    I1 + I2p1 p2 = m0(2 2) = m0(cos2 sin2 ) =

    2I1I2

    I1 + I2cos 2

    Kelvin Wagner, University of Colorado Physical Optics 2011 86

    3D Polarization StateInterference of p-polarization

    2 = 60o

    XY

    Z

    Y

    60o

    2 = 5o

    XY

    Z

    Y

    Interference between in-plane p-polarization produces only in-plane components, butfor large angles the interference will contain z-components, with phase that variesperriodically across the interference fringe. For small angles these z-components canbe neglected.

    Kelvin Wagner, University of Colorado Physical Optics 2011 87

    3-D Polarization StateInterference of Orthogonal Circular

    Polarization

    XY

    Z

    Y

    /2/4

    0/4

    /2

    = 0:514m at 514nm

    XY

    Z

    Y

    /2

    /2

    0/4

    /4

    = 5:9m at 514nm

    One period of the spatially varying polarization state of total recording field shownin three-dimensional perspective with projections onto the XY and Y Z planes fororthogonal circular recording beams with beam ratio mo = 4, where (a) is a large anglecase with 2 = 60o and (b) is a small angle case with 2 = 5o.

    Kelvin Wagner, University of Colorado Physical Optics 2011 88

  • Orthogonal Linear Polarization Interference

    Vertical Horizontal

    om = 1

    (a)

    I rx Isy

    (b)

    left circular

    right circularver

    tical

    (refe

    rence

    )

    horiz

    ontal

    (sign

    al)

    Vertical Horizontal

    om = 4

    I rx Isy

    (c)

    left circular

    right circular

    ver

    tical

    (refe

    rence

    )

    horiz

    ontal

    (sign

    al)

    (d)

    Spatially varying polarization state of the total amplitude field formed by two orthogo-nal linearly polarized beams incident o n the input face of the DPOM. (a). Beam ratiomo = 1. (b). Corresponding Poincares sphere representati on to (a). (c). Beam ratiomo = 4. (d). Corresponding Poincares sphere representati on to (c).

    Kelvin Wagner, University of Colorado Physical Optics 2011 89

    Orthogonal Circular Polarization Interference

    m = 1o

    RightCircular

    LeftCircular

    (a)I r I s

    right circular (reference)

    left circular (signal)

    ver

    tical

    horiz

    ontal

    (b)

    RightCircular

    LeftCircular

    om = 4

    I r I s

    (c)

    right circular (reference)

    left circular (signal)

    ver

    tical

    horiz

    ontal

    (d)

    Spatially varying polarization state of the total amplitude field formed by two orthog-onal circular polarized beams. (a). Beam ratio mo = 1. (b). Corresponding Poincaressphere representati on to (a). (c). Beam ratio mo = 4. (d). Corresponding Poincaressphere representation to (c).

    Kelvin Wagner, University of Colorado Physical Optics 2011 90


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