Intro Optics - PPT v2part 04

8/23/2019 Intro Optics - PPT v2part 04

1/56

Michelson Interferometer 0Michelson InterferometerThe interferometer experiment of Michelson and Morley played a

crucial role in Einsteins development of the Special Theory of

Relativity.


2/56

Michelson Interferometer 1Michelson InterferometerA wavefront from the source is split into two equal amplitude

components which are recombined at the detector.

Path lengths (1) S M1 D and (2) S M2 D are generally

different. One mirror is moveable.


3/56

Michelson Interferometer 2Michelson InterferometerThe operation of the device can be understood in terms of

interference in a dielectric film. To see this we can conceptually

straighten out the device by bringing the 2 arms into alignment.

The path length difference is d, the mirror separation. As in the

film, circular Haidinger fringes are mapped out on the detector.


4/56

Michelson Interferometer 3Michelson Interferometer

Michelson

equivalent:

The optical path

difference (OPD)

between fields

along paths (1)

and (2) is:

giving the phase difference:


5/56


The action of the beam

splitter generally results

in an additional phasedifference of betweenthe fields in paths (1)

and (2) . Field (2)

undergoes an external

reflection while field (1)undergoes an internal

reflection at the beam

splitter. Thus the overall

phase difference is:


6/56

Michelson Interferometer 5Michelson InterferometerRecombined beams at the detector give irradiance determined by

the Two Beam Interference Formula (equal amplitude):

with

The condition for a dark fringe (irradiance minimum) is:

which can be written as:


7/56

Michelson Interferometer 6Michelson InterferometerThe irradiance at the detector is usually written in terms of the

quantity , which is the OPD between the two arms:

where


8/56

Michelson Interferometer 7Michelson InterferometerBehaviour of the fringes.

for a given m, cosm decreases as d increases m increases as d increases. Low order fringes move outward to higher angles with

increasing d.

The dark fringe condition (mth order fringe):


9/56


for neighbouring fringes (m = 1) , decreases as dincreases

field of view becomes congested with more fringes as d

increases.

The angular spacing of the fringes:


10/56



11/56



12/56

Michelson Interferometer 10aMichelson Interferometer

19.994/10094

18.295/10095

16.396/10096

14.197/10097

11.598/10098

8.199 /10099

0110050

25.827 /3027

21.028 /3028

14.829 /3029

013015

9000

601 /21

012

9000

011 /2

9000 /4

anyundefined00

(degrees)cosmmd

m = 2dcosm


13/56

Michelson Interferometer 11Michelson InterferometerFor misaligned mirrors, the Michelson interferometer is

equivalent to the dielectric wedge geometry.

This gives the Fringes of Equal Thickness (localized at

mirror M1). The central fringe is = 0.


14/56


For misaligned mirrors, the Michelson interferometer is

equivalent to the dielectric wedge geometry.

This gives the Fringes of Equal Thickness (localized at

mirror M1). The central fringe is = 0.


15/56

Michelson Interferometer: Apps1

Michelson Interferometer: Applications

Measuring the Wavelength of Light

We set up the

interferometer to monitor

the central fringe ( =0).

This can be done using a

lens and an aperture in the

lens focal plane to let only

=0 light into the detector.


16/56



Measuring the Wavelength of Light

We set up the

interferometer to monitor

the central fringe ( =0).

It can also be done using a

telescope to visually align

the =0 fringe with thetelescope crosshairs. This

will be the technique used

in the lab.


17/56



The MI is set up to view the central fringe.

The dark fringe condition is:

The order of the central fringe is:

If d is changed by d then m changes by m:


18/56



An increase in d from d d + dgives an increase in m from m m + m.

Thus, m fringes are counted at the detector as d d + d .By counting m fringes for known d we get:

This can be a very accurate technique, especially when a largenumber of fringes are counted for an accurately measured mirror

displacement d .


19/56



The technique can be inverted to find d in terms of wavelength.This is the basis of an atomic standard of length when a precisely

known wavelength of an atomic spectral line is used.

Interferometer used as

length standard at the

National MetrologyInstitute of Japan.


20/56



Refractive Index of a Gas

In the arrangement shown, a

gas cell is placed in one arm of

the MI and initially evacuated

(nvac 1). Gas is then leakedinto the cell until a desired

pressure is reached. The

corresponding refractive index

is ng (at that pressure).

As gas leaks into the cell, the

order of the central fringe

changes.


21/56



Assume the central fringe is initially dark (the MI can be set up thisway). The (central) dark fringe condition is:

As gas pressure increases, n will change in the cell and there will

be a corresponding change in m:

m is the number of central fringes counted with the change inOPD of(nd).

Change in OPD

between the arms.


22/56



The change in OPD:

Evacuated

CellFilled

Cell


23/56



Refractive Index of a Glass Plate

Similar type of measurement

to the gas cell experiment.

Here, we begin by setting upthe condition of Zero OPD.

How? We look for the

presence of white light

fringes (more later).


24/56

Michelson Interferometer: Apps9a



Similar type of measurement

to the gas cell experiment.

Here, we begin by setting upthe condition of Zero OPD.

How? We look for the

presence of white light

fringes (more later).

Then insert the glass plate.

The change in OPD with the

glass plate inserted:


25/56




Now, with the glass plate

inserted, the mirror

separation is adjusted toreestablish the white light

fringe observation

condition. If the change in

mirror separation required to

re-capture the white lightfringes is d, then:

and the index of refraction can be determined if the thickness of the

plate is known.


26/56



Precision Spectroscopy: Resolving the Sodium Doublet

The MI can be used to measure the wavelength separation between

(resolve) two closely spaced spectral lines (eg the NaDoublet).


27/56



Each wavelength component gives rise to its own fringe pattern(with different fringe spacing). At some mirror separations, d, dark

fringes from 01 will overlap bright fringes from 02 and thecombined fringe pattern will wash out ie no fringes will be visible.

The irradiance at the detector due to each wavelength component:

01 :

02 :

with

The total irradiance at the detector:


28/56



Assuming I01 I02 I0 (where I0 is the total irradiance from bothwavelength components in each arm) the irradiance at the detector

becomes:


29/56



We rewrite this as:

The cosAx term is rapidly

varying (generating the

fringes) while the cosBxterm is slowly varying.

I

/

2I0

x


30/56



We rewrite this as:



fringes) while the cosBxterm is slowly varying (the

fringe envelope).

I

/

2I0

x


31/56



We rewrite this as:

Bx = / 2

(m=0)



fringes) while the cosBxterm is slowly varying (the

fringe envelope).

The fringes vanish when

cosBx = 0

ie

Bx = (2m+1) / 2

x

I

/

2I0


32/56



Thus the fringes vanish (wash out) when:

We can calculate the difference in wavenumberk0 of the 2spectral lines if we measure the mirror separation dm correspondingto the mth order in which the fringes disappear:


33/56



Its sometimes more meaningful to find the difference in wavelength0 rather than difference in wavenumberk0 of the 2 spectral lines.

We can relate the two differences by:

Where 0 is taken to be the mean value:

So:

or for normal viewing ( = 0).


34/56



A better technique (as you will use

in the lab) involves measuring thechange in mirror separation dmrequired to go from one fringe

disappearance (mth order) to the

next ((m+1)th order). For this

measurement its unnecessary toknow the order, m, to determine the

wavelength difference, 0 .dm

dm dm+1

d


35/56



The change in mirror separation dm required to go from one fringedisappearance (mth order) to the next ((m+1)th order) gives 0 :


36/56



White Light Fringes

For the monochromatic (single wavelength) source:

Spectrum Fringe Pattern: I()


37/56



For the doublet (two wavelength) source:



38/56



For the white light source:


f(k)


39/56



An approximation to a white light spectrum: Model Spectrum

Model Spectrum

Here:

Isource is the total (integrated over all

wavelengths) intensity from the

source. I0 is the integrated intensity

in one arm of the MI.


40/56



By evaluating the integral

we can show that (details follow):


41/56



or:

where the sinc function is defined by sinc(x) = sinx / x


42/56



For the white light model spectrum source:

Spectrum Fringe Pattern: I()Fringe pattern looks somewhat like the real deal!


43/56



Details: Evaluation of the model white light fringe pattern.


44/56



Details (cont.)


45/56



Optical Coherence Tomography (OCT)

Optical coherence

tomography is a recently

developed, noninvasivetechnique for imaging

subsurface tissue structure

with micrometer-scale (10-6

m) resolution. Depths of 12mm can be imaged in turbid

tissues such as skin or

arteries; greater depths are

possible in transparent

tissues such as the eye.

OCT image of the optic nerve head.


46/56



Optical Coherence Tomography

SLD: superluminescent Diode (white light source) REF: (reference) mirror

SMP: sample BS: beamsplitter CO-CAM: camera objective - camera

The device is a Michelson interferometer used with white light!


47/56



Light that has been back-reflected from the tissue sample (index-of-

refraction mismatches) and light from the reference arm recombine.

Because the source is a broadband (white light) source, only lightwhich has traveled very close to the same optical path length in the

reference and tissue arms will generate interference fringes. The

fringes are detected at the camera. By changing the length of the

reference arm, reflection sites at various depths in the tissue canbe sampled.

Optical Coherence Tomography


48/56



Variations of the Michelson Interferometer.


49/56



Twyman Green variation:

Twyman Green interferometer

uses a point source and

collimating lens as opposed to

the extended source in the

Michelson. As shown it isconfigured to test a lens.

Lens under test

Convex reflector


50/56



Twyman Green variation:

Fringe patterns for an aberrated lens:

Spherical Coma Astigmatism

Aberration


51/56



Mach Zehnder Interferometer (Fusion Plasma Diagnostics)


52/56



Linnik Interference Microscope.

Used for mapping surface topography.

Mi h l I f A li i


53/56



Linnik Interference Microscope.

linnik.gif

Mi h l I t f t A li ti


54/56



LIGO: Laser Interferometer Gravitational Wave Observatory

What Will LIGO Observe?

Gravitational waves triggered bycosmic events should cause

specific displacements resulting

in unique interference patterns.

Mi h l I t f t A li ti


55/56




LIGO CalTech prototype (40m arms).



56/56




LIGO Hanford Observatory, Washington State.

Documents

Intro Optics - PPT v2part 04