# interferometers and interferometry

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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

• 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

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) ]

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

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