11-30 Two Beam Interferoemters

8/6/2019 11-30 Two Beam Interferoemters

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Michelson Interferometer

Sagnac Interferometer

Fizeau Interferometer

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L1

L2

=4(n1L1-n2L2)/

Measuring the intensity of theoutput allows us to infer the values

of physical parameters n, L or

I0

Iout


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From 1980-1983 the meter was defined as1579800.299 wavelengths of light from a HeNelaser. How could you reproduce the standard

meter in a lab?

Scan one mirror as you count off twice thismany fringes. The distance it travels would be

one meter


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The output field varies depending on wavelength and on the path lengthdifference

giving an output intensity (for r2=t2=1/2) of

if one of the mirrors moves at constant velocity, such that

The output signal will be modulated at a frequency

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With the expression

the wavelength of a source can bemeasured by comparing the modulationfrequency for that source to themodulation frequency of a knownreference

If N=L/ is the number ofinterference fringes that are countedthe measurement error / is 1/Ngiving a resolving power of

Output intensity as a function of time

Schematic of a scanning Michelson interferometersource: Andrews and Demidov

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L1

L2

=4(n1L1-n2L2)/

Measuring the intensity of theoutput allows us to infer the values

of physical parameters n, L or

I0

Iout

vacuum chamber


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How can you measure the index ofrefraction of air?

L1

L2

=4(n1L1-n2L2)/

I0

Iout

vacuum chamber

count the fringes as air is let back into a vacuumchamber of known length in one arm


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What do we mean by count the fringes?

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04 -7500 -5000 -2500 0 2500 5000 7500 1!1

-5000

-2500

2500

5000

fringe pattern of analigned Michelson withan arm length mismatch

ofL

fringe pattern of analigned Michelson withan arm length mismatch

ofL+/10

04 -7500 -5000 -2500 0 2500 5000 7500 1!1

-5000

-2500

2500

5000


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How accurately can we count fringes?

Shot noise produces noise that causes the intensity to fluctuate by

where N=Pt/h is the number of photons detected in our measurement of

time t. This is related to the phase shift on the side of a fringe by

So the minimum detectable phase shift is

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Veff =v

vI= I/

N

0

I

dI

d= I

min =1N

=

h

Pt


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What is the smallest displacement of a Michelson mirror that can bemeasured by a 500 mW Nd:YAG laser with a wavelength of=1064nm in aone second measurement?

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min =1

N=

h

Pt

=

hc

Pt

= (6.63 1034 J s)(3 108 m/s)(1064 109 m)(500 103 W)(1 s)

= 6 1010 radfrom =2(2L)/ for a Michelson interferometer

Lmin=min/4=5x10-17 m


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A detector with a finite size aperture wil tend toaverage out the fringes that are incident on thedetector. The effective visibility is

and the actual detectable phase shift is

act=min/Veff

one could minimize the detector size to avoid thisproblem, but that would limit the detected power andhence decrease min. Alternatively if theinterferometer is well aligned, the fringes becomecourser than the detector 13

Veff = vmax

vmin

vmax + vmin


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What does the output interferencepattern look like when theinterferometer is aligned? When it

is misaligned?

L1

L2

=4(n1L1-n2L2)/

I0

Iout


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Two counter-propagatingbeams interfere. Sincebeams travel the samepath (in opposite

directions) this devicehas very good commonmode noise rejection, andis primarily sensitive topath length changes at

high frequency

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Consider the effect ofrotation on a squareSagnac interferometer

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transit time is tt=4L/c, additional path lengthdue to rotation during that time is LLtt,so =16L2/c or =16A/, and is avery sensitive measure of rotation

L

note: to treat this problem properly requires a full relativistic treatment not considered here


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An air wedge between two glassplates to generate fringes that have aspacing proportional to thewavelength

To prevent aliasing on the m-elementdetector array we must have thenumber of fringes N=2d/ less thanm/2 giving a limit on d


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Many configuration of amplitude splittinginterferometers exist that allow physicalquantities to be measured with extremesensitivity

Even if all technical sources of noise can beeliminated, shot noise will still exist and limitthe sensitivity of any optical measurement

Alignment, power, and measurement time allaffect the sensitivity of measurements madewith an interferometer

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11-30 Two Beam Interferoemters