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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
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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
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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
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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
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Source with random phase jumps at equalintervals
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Source with random phase jumps at randomintervals
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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
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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
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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
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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
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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
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Spherical Aberration: 1 waveStrehl Ratio .09
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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
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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
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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
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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
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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
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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
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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
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3rd order Astigmatism : 1 wave
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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
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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
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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
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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
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Multiple Reflection Artifacts in Beamsplitters
Michelson Interferometer MachZehnder Interferometer
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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
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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