Pupils, windows, aberrations

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  • Aberration Theory

    GabrielPopescu

    UniversityofIllinoisatUrbanaChampaigny p gBeckmanInstitute

    Quantitative Light Imaging Laboratory

    Electrical and Computer Engineering, UIUCPrinciples of Optical Imaging

    QuantitativeLightImagingLaboratoryhttp://light.ece.uiuc.edu

  • 1. Pupils and Windows

    ECE460 OpticalImaging

    Consideraobjectimagedbythesystembelow:

    1.PupilsandWindows

    j g y y

    Imaging

    PPC

    B B

    Imaging System

    B

    A A

    B

    ! The dotted lines do not make it through the system i e they!Thedottedlinesdonotmakeitthroughthesystem,i.e,theyareblockedsomewhere

    Aberrations 2

  • 1. Pupils and Windows

    ECE460 OpticalImaging

    Twomaintypeoflimitations:

    1.PupilsandWindows

    yp

    a) Solidangle fromthepointonaxisislimittedbytheENTRANCEPUPIL

    b) fieldofview(i.eextentoftheobject)limittedbyENTRANCE WINDOWENTRANCEWINDOW

    3Aberrations

  • 1. Pupils and Windows

    ECE460 OpticalImaging

    a) Pupils

    1.PupilsandWindows

    ) p

    xblocked

    OA

    B

    OAA

    Entrance Pupil image of aperture stop in the object spaceAPERTURE STOP

    EntrancePupil imageofaperturestopintheobject space

    4Aberrations

  • 1. Pupils and Windows

    ECE460 OpticalImaging

    1.PupilsandWindows

    aperture stopmaximum

    Bsolid angle

    A

    Exit Pupil image of aperture stop in the image space

    entrancepupil

    exitpupil

    ExitPupil imageofaperturestopintheimage space

    5Aberrations

  • 1. Pupils and Windows

    ECE460 OpticalImaging

    EntrancePupil:

    1.PupilsandWindows

    p

    Limitssolidangle

    Thus,limitstheamountoflight,i.ebrightness ofimage

    Anglefromobjectisproportionaltospatialfrequency

    So:

    Adjustingthepupilofoureyes,weadjustbrigthness andresolution

    High brightness = high resolution but may induceHighbrightness highresolution,butmayinducesaturationandaberations

    CommonissueinSLRphotography

    6Aberrations

  • 1. Pupils and Windows

    ECE460 OpticalImaging

    Note:Tofindentrancepupil,imageall opticalelementsin

    1.PupilsandWindows

    p p , g pobjectplaneandpicktheonethatsubtendthesmallestanglefromobject

    entrance pupil

    O CO.C

    7Aberrations

  • 1. Pupils and Windows

    ECE460 OpticalImaging

    b)Windows

    1.PupilsandWindows

    )

    Limithesolidanglemadebyraysatthecenteroftheentrancepupil(chief rays)withOA

    fi ld

    Bexit pupilentrance

    pupil

    field of view

    B

    EntrancePupil=imageoffieldstopintheobjectspace

    Bblocked field stop

    chief ray

    ExitPupil=imageofthefieldstopintheimagespace

    8Aberrations

  • 1. Pupils and Windows

    ECE460 OpticalImaging

    Limitsthefieldofview

    Note:

    1.PupilsandWindows

    Note:

    SpaceandangleareFourierrelated

    Itisalwaysacompromisebetweenfieldofviewandsolidangle(NA)

    Tofindtheentrancewindow:imageallcomponentsinobject space and pick the one subtending the smallest angleobjectspace,andpicktheonesubtendingthesmallestanglefromthecenteroftheentrancepupil.

    entrance pupil

    A.

    9

    A

    entrance windowAberrations

  • 2. Frequency analysis of coherent imaging

    ECE460 OpticalImaging

    2.Frequencyanalysisofcoherentimagingentrance pupil

    exitpupil ( , )p

    IMAGINGSYSTEM

    A

    B

    A

    Recallthattheimagefieldequalstheobjectfieldconvolvedwiththeimpulseresponseofthesystem(Eq4.4)

    ( ' ') ( )U x y U x y ( )h x y 1 a)( ', ') ( , )U x y U x y ( , )h x y2 [ ]( , ) [ ( , )] ( , ) i x yh x y H H e d d

    )

    1 b)

    H=transferfunction(coherent)

    10

    Aberrations

  • 2. Frequency analysis of coherent imaging

    ECE460 OpticalImaging

    2.Frequencyanalysisofcoherentimaging

    0 0( , )p x y = exit pupil Note:

    p p0 0; ( , ) ( , )z z

    x y H P z z

    y0

    10

    y0

    A

    0 0( , )p x y

    R x0A

    Example of a pupilB

    Z

    11Aberrations

  • 2. Frequency analysis of coherent imaging

    ECE460 OpticalImaging

    Mostcommonpupilfunction:

    2.Frequencyanalysisofcoherentimaging

    p p

    0 0( , )p x y 1,0

    2 2 20 0x y R

    otherwise(2)

    So:ImpulseresponsehistheFraunhofferdiffractionpatternofP!

    0, otherwise

    Thisisadiffractionlimittedinstrument,i.e.Thebestwecandoinpractice

    12Aberrations

  • 3. Incoherent Imaging

    ECE460 OpticalImaging

    Iftheilluminatingfieldisspatiallyincoherent,i.e:

    3.IncoherentImaging

    *( ) ( ) ( ) ( )U U I (3)

    Thesystembecomeslinearinintensities:

    1 1 2 2 1 1 1 2 1 2( , ; ) ( , ; ) ( , ) ( ; )U x y t U x y t I x y x x y y (3)

    ( ', ') ( , )I x y I x y 2| ( , ) |h x y (4) 2 *| |h h h [ ]h h

    Theintensityimpulseresponse:|h|2!

    Incoherenttransferfunction:*

    [ ]

    [ ]

    h h

    h h

    2 * *1( , ) [| ( , ) | ] [| ] ( , ) ( , )H h x y h h H H

    = autocorrelation of coherent transfer function

    (5)

    Opticaltransferfunction:13

    1 1( , ) / (0,0)OTF H H (5)Aberrations

  • 4. Effects of Aberrations

    ECE460 OpticalImaging

    Sofar,weassumedthatpointsareimagedintopoints,

    i.e perfect imaging

    4.EffectsofAberrations

    i.eperfectimaging

    i.ediffractionlimittedsystem

    i.epointsourceobjectgeneratesasphericalwavefrontatexit

    [ ]Sin

    pupil

    Inreality,theoutgoingwavefrontisnonspherical,i.eaberrated

    actual wavefront

    ideal wavefront

    wavefrontw(x,y)

    14

    wavefront (reference) w(x,y) = path-length error

    Aberrations

  • 4. Effects of Aberrations

    ECE460 OpticalImaging

    Alltheeffectsofwavefronterrors,i.eaberrationscanbeaccountedforbygeneralizingthepupilfunctions:

    4.EffectsofAberrations

    y g g p p

    Note:

    ( , )'( , ) ( , ) ikW x yP x y P x y e (6)

    W(x,y)canbepositiveornegative

    Wisthelocaldifferencebetweentheactual wavefrontandthe reference onethereference one

    handHfollowfromEq(6):

    a) Coherent illumination:a)Coherentillumination:( , )'( , ) '( , )

    ( ) [ ( )]

    ikW x yH P z z Peh x y H

    (7)

    15

    ( , ) [ ( , )]h x y H

    Aberrations

  • 4. Effects of Aberrations

    ECE460 OpticalImaging

    Wcanhavesevereeffectsonthefinalimage

    Exercise: Calculate h for

    4.EffectsofAberrations

    2 2( )k W a Exercise:Calculatehfor

    b)IncoherentIllumination:

    ( )k W a

    RecallEq.5:*

    1 '( , ) ( , ) ( , )H H H []*

    Expressingtheautocorrelationintegralforapupilfunction

    ( , )( , ) ikW z zP z z e (8)

    ( ) 1P i id( , ) 1,( , ) 0,

    P x y insideP x y outside

    , ,2 2 2 2'( )x y x yik W x y W x y

    H d d

    16

    2 2 2 21

    ( , )'( , )

    AH e dxdy

    (9)

    Aberrations

  • 4. Effects of Aberrations

    ECE460 OpticalImaging

    4.EffectsofAberrations

    x 0P(x,y)

    Hz

    y

    1 0

    z

    Area ( , )A

    2 2Low Pass

    Note:

    Ab ti l d hi h f f t

    2 21 1'( , ) ( , )H H (10)

    Aberrationscanonlyreducehighfreqofpowerspectrum

    Notethatk=2/ wavelengthdependencecanhaveanefect,too!Importantinbroadspectrumimaging(e.g.white, p p g g ( glight).

    17Aberrations

  • 5. Chromatic Aberrations

    ECE460 OpticalImaging

    Recallthedispersioncurvefromchapter1.8(Lorentzmodel):

    5.ChromaticAberrations

    n()

    1

    n ()

    n()

    n refractiveindex

    b ti

    0

    n absorption

    We assume n 0 i e our optical components are higly Weassumen 0,i.eouropticalcomponentsarehiglytransparent

    18Aberrations

  • 5. Chromatic Aberrations

    ECE460 OpticalImaging

    Consideralensmadeofglasswithn()

    5.ChromaticAberrations

    R

    R1>0

    R2

  • 5. Chromatic Aberrations

    ECE460 OpticalImaging

    ThinLens:t=0

    5.ChromaticAberrations

    1

    1 1 1

    1Cf

    1 2

    1 1 1( 1)nf R R

    (12)

    If|R1|=|R2|

    1 2( 1)n (13)

    f R (13)

    20Aberrations

  • 5. Chromatic Aberrations

    ECE460 OpticalImaging

    Thedispersioneffectisveryclear:

    5.ChromaticAberrations

    focaldistancevarieswithcolor! (14)( )2[ ( ) 1]

    Rfn

    DifferentiateEq.14:

    1 2( 1)nd d (1 )( )1 2

    d df R

    df dn

    (15)

    (16)2

    fd R df

    22df f dn

    (16)

    21Aberrations

    f fd R d

    (17)

  • 5. Chromatic Aberrations

    ECE460 OpticalImaging

    5.ChromaticAberrations

    22df f dn 0 normal dispersiondn

    Whatisthenegativesigntelling?

    d R d 0 normal dispersiond

    IR Red Green Blue UV

    white lightmarginal ray

    longitudinal chromatic aberration

    OAg y

    B Y R

    22Aberrations

  • 5. Chromatic Aberrations

    ECE460 OpticalImaging

    5.ChromaticAberrations

    longitudinalR

    OA

    longitudinal chromatic aberration

    RGB

    Mostcommoncorrection:theachromaticdoublet

    S d i h f t l iti ti Sandwichoftwolenses,onepositive,onenegative,whichcorrectsthechromaticaberrationat2colors(R&B)

    23Aberrations

  • 5. Chromatic Aberrations

    ECE460 OpticalImaging

    5.ChromaticAberrations

    marginal

    n1 n2Y RB

    Y

    F thi l l k d l

    YRB

    Forthin,closelypackedlenses:

    1 1 1f f f (18)

    24Aberrations

    1 2f f f( )

  • 5. Chromatic Aberrations

    ECE460 OpticalImaging

    Forthin,closelypackedlenses