interferometers history

1

INTERFEROMETERSHISTORY

SOLO HERMELINUpdated: 6.01.11

4.01.15http://www.solohermelin.com

http://www.solohermelin.com/

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InterferenceSOLO

Table of Content

Introduction

Haidinger Fringes

Fizeau Experiment

Jamin’s Interferometer

Fizeau Experiment in Moving Media and Fizeau Interferometer

Foucault Experiment

Michelson Interferometer

Mach-Zehnder Interferometer

Rayleigh’s Interferometer

Sirks-Pringsheim Interferometer

Fabry – Perot Interferometer

Sagnac Effect

Twyman-Green Interferometer

Michelson’s Experiments

Dyson’s Interferometer

Hanbury-Brown and Twiss InterferometerGires-Tournois Etalon

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InterferenceSOLO

Table of Content (continue – 1)

Interference of Two Monochromatic Waves

Two Basic Classes of Interferometers

Fresnel’s Double Mirror (1819*)

Fresnel’s Double Prism

Lloyd’s Mirror Interferometer

Young’s ExperimentDivision of Wavefront

Optical Reflected Path Length Difference: Parallel Interfaces

Amplitude Split InterferometersStokes Treatment of Reflection and Refraction

Optical Transmitted Path Length Difference: Parallel Interfaces

Haidinger Interference Fringes

Interference of Many Monochromatic Waves

Gas Refrectometer

References

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InterferenceSOLO

Introduction

Interference is the superposition of two or more waves producing a resultant disturbance that is the sum of the overlapping wave contribution.

More work must be done to complete this presentation

5

SOLO Diffraction

The Grimaldi’s description of diffraction was published in1665 , two years after his death: “Physico-Mathesis de lumine,Coloribus et iride”

Francesco M. Grimaldi, S.J. (1613 – 1663) professor of mathematics and physics at theJesuit college in Bolognia discovered the diffraction of light and gave it the namediffraction, which means “breaking up”.

http://www.faculty.fairfield.edu/jmac/sj/scientists/grimaldi.htm

“When the light is incident on a smooth white surface it will show an illuminated base IK notable greater than the rays would make which are transmitted in straight lines through the two holes. This is proved as often as the experiment is trayed by observing how great the base IK is in fact and deducing by calculation how great the base NO ought to bewhich is formed by the direct rays. Furter it should not beomitted that the illuminated base IK appears in the middlesuffused with pure light, and either extremity its light iscolored.”

Single SlitDiffraction

Double SlitDiffraction

http://en.wikipedia.org/wiki/Francesco_Maria_Grimaldi

1665

6

SOLO Microscope

Robert Hooke (1635-1703) work in microscopy is described in Micrographia published in 1665. Contains investigations of the colours of thin plates of mica, a theory of light as a transverse vibration motion (in 1672).

1665

Robert Hooke (1635-1703)

Robert Hooke’s compoundmicroscope: on the leftthe illumination device

(an oil lamp), onthe right the microscope.

Robert Hooke reports in Micrographia the discovery of the rings of light formed by a layer of air between two glass plates, first observed by Robert Boyle. In the same work he gives the matching-wave-frontderivation of reflection and refraction. The waves travel through aeter.

Robert Hooke also assumed that the white light is a simple disturbance and colors are complex distortion of the white light. This theory was refuted later by Newton.

7

Optics SOLO

Newton published “Opticks”

1704

In this book he addresses:• mirror telescope• theory of colors• theory of white lite components• colors of the rainbow • Newton’ s rings • polarization• diffraction • light corpuscular theory

Newton threw the weight of his authority on thecorpuscular theory. This conviction was due to thefact that light travels in straight lines, and none of the waves that he knew possessed this property.

Newton’s authority lasted for one hundred years,and diffraction results of Grimaldi (1665) and Hooke (1672), and the view of Huygens (1678) were overlooked.

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InterferenceSOLO

Newton Fringes

ColllimatorLens

Beam-splitter Point

Source

ViewingScreen

Opticalflat

Circularfringes

BlackSurface

9

OpticsSOLO

History (continue)

In 1801 Thomas Young uses constructive and destructive interference of waves to explain the Newton’s rings.

Thomas Young1773-1829

1801 - 1803

In 1803 Thomas Young explains the fringes at the edges of shadows using the wave theory of light. But, the fact that was belived that the light waves are longitudinal, mad difficult the explanation of double refraction in certain crystals.

10

POLARIZATIONSOLO

History

In 1802 William Hyde Wollaston discovered that the sun spectrumis composed by a number of dark lines, but the interpretation of thisphenomenon was done by Fraunhofer in 1814. Wollaston developed In 1802 the refractometer, an instrument used to measure the refractiveindex. The refractometer wa used by Wollaston to verify the laws of double refraction in Iceland spar, on which he wrote a treatise.

Wollaston Prism

In 1807 William Hyde Wollaston developed the four-sided Wollaston prism, usedin microscopy.

1802 - 1807

a, 08/20/2005

M.V.Klein, T.Furtak,"Optics", pp.34-35E. Hecht, A. Zajac,"Optics",pp.8, pp.225-226

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SOLO

History

In 1813 Joseph Fraunhofer rediscovered William Hyde Wollaston’s dark lines in the solar system, which are known as Fraunhofer’s lines.He began a systematic measurement of the wavelengths of the solar Spectrum, by mapping 570 lines.

Diffraction

http://www.musoptin.com/spektro1.html

1813

Fraunhofer Telescope.

Fraunhofer placed a narrow slit in front of a prism and viewed the spectrum of lightpassing through this combination with a small telescope eypiece. By this technique he was able to investigate the spectrum bit by bit, color by color.

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SOLO

Between 1805 and 1815 Laplace, Biot and (in part) Malus created an elaborate mathematical theory of light, based on the notion that light rays are streams of particles that interact with the particles of matter by short range forces. By suitably modifying Newton’s original emission theory of light and applying superior mathematical methods, they were able to explain most of the known optical phenomena, including the effect of double refraction (Laplace 1808) which had been the focus of Huyghen’s work.

Diffractionhttp://microscopy.fsu.edu/optics/timeline/people/gregory.html

http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html

Pierre-Simon Laplace(1749-1827)

1805 - 1815

In 1817, expecting to soon celebrate the final triumph of their neo-Newtonian optics,

Laplace and Biot arranged for the physics prize of the French Academy of Science to be proposed for the best work on theme of diffraction – the apparent bending of light

rays at the boundaries between different media.”

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SOLO

In 1818 August Fresnel supported by his friend André-Marie Ampère submitted to the French Academy a thesis in which he explained the diffraction by enriching the Huyghens’ conception of propagation of light by taking in account of the distinct phases within each wavelength and the interaction (interference) between different phases at each locus of the propagation process.

Diffractionhttp://microscopy.fsu.edu/optics/timeline/people/gregory.html http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html

André-Marie Ampère(1775-1836)

Dominique François Jean Arago1786-1853

Siméon Denis Poisson1781-1840

Pierre-Simon Laplace(1749-1827)

Joseph Louis Guy-Lussac1778-1850

JudgingCommittee

ofFrench

Academy

1818

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SOLODiffraction

http://microscopy.fsu.edu/optics/timeline/people/gregory.html http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html


Siméon Denis Poisson a French Academy member rise the objection that if the Fresnel construction is valid a bright spot would have to appear in the middle of the shadow cast by a spherical or disc-shaped object, when illuminated, and this is absurd.

Soon after the meeting, Dominique Francois Arago, one of the judges for the Academy competition, did the experiment and there was the bright spot in the middle of the shadow. Fresnel was awarded the prize in the competition.

Siméon Denis Poisson1781-1840

Poisson’s or Arago’s Spot

1818

15

POLARIZATION

Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that tworays polarized at right angles to each other never interface.

SOLO

History (continue)


Augustin Jean Fresnel

1788-1827

Arago relayed to Thomas Young in London the resultsof the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillationsin the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young

1773-1829

1816 - 1817

longitudinalwaves

transversalwaves

a, 08/20/2005

M. Born, E. Wolf,"Principles of Optics", pp. xxiiiM.V. Klein, T. Furtak,"Optics", pp.35

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Diffraction SOLO

History of Diffraction


1788-1827

In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves.

1818

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Diffraction SOLO


1788-1827

In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves.

P

0P

Q1x

0x1y

0y η

ξ

Fr

Sr

ρr

O

'θ θ

Screen

Image

plane

Source

plane

0O

1O

Sn1

Σ

Σ - Screen Aperture

Sn1 - normal to Screen

From a source P0 at a distance from a aperture a spherical wavelet propagates toward the aperture: ( ) ( )Srktj

S

sourceQ e

r

AtU −= '' ω

According to Huygens Principle second wavelets will start at the aperture and will add at the image point P.

( ) ( ) ( )( ) ( ) ( )( )∫∫Σ

++−

Σ

+−− == drerr

AKdre

r

UKtU rrktj

S

sourcerkttjQP

S 2/2/' ',', πωπω θθθθ

where: ( )',θθK obliquity or inclination factor ( ) ( )SSS nrnr 11cos&11cos' 11 ⋅=⋅= −− θθ

( )( )

======0',0

max0',0

πθθθθ

K

K Obliquity factor and π/2 phase were introduced by Fresnel to explain experiences results.

Fresnel Diffraction Formula

Fresnel took in consideration the phase of each wavelet to obtain:

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SOLO

History

In 1821 Joseph Fraunhofer build the first diffraction grating, made up of 260 close parallel wires. Latter he built a diffractiongrating using 10,000 parallel lines per inch.

Diffraction

Utzshneider, Fraunhfer, Reichenbach, Mertz

http://www.musoptin.com/fraunhofer.html

1821 - 1823

In 1823 Fraunhofer published his theory of diffraction.

http://micro.magnet.fsu.edu/optics/timeline/people/fraunhofer.html

19

SOLO ELECTROMAGNETICS

Conical Refraction on the Optical Axis (continue – 2) Take a biaxial crystal and cut it so that two parallel faces

are perpendicular to the Optical Axis. If a monochromaticunpolarized light is normal to one of the crystal faces, theenergy will spread out in the plate in a hollow cone, thecone of internal conical refraction.

When the light exits the crystal the energy and wave directions coincide, and the light will form a hollow cylinder.

This phenomenon was predicted by William Rowan Hamiltonin 1832 and confirmed experimentally by Humphrey Lloyd, a year later (Born & Wolf).

Because it is no easy to obtain an accurate parallel beam ofmonochromatic light on obtained two bright circles

(Born & Wolf).

1832William Rowan

Hamilton(1805-1855)

Humphrey Lloyd(1800-1881)

a, 06/10/2006

See treatment in 1. Born & Wolf, "Principle of Optics", 6th Ed., pp.686-6902. M.V. Klein,"Optics", 1970, pp.609

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SOLO

Airy Rings

In 1835, Sir George Biddell Airy, developed the formula for diffraction pattern, of an image of a point source in an aberration-free optical system, using the wave theory.

E. Hecht, “Optics”

Diffraction 1835

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InterferenceSOLO

Haidinger Fringes1846

Wilhelm Karl, Ritter von Haidinger

1795 - 1871

Lens

Beam-splitter

ExtendedSources

ViewingScreen

Dielectricfilm

Blackbackground

Circularfringes

Haidinger Fringes are the type of interference pattern that results with an extended source where partial reflectionsoccur from a plane-parallel dielectric slab.

Return to Table of Content

http://en.wikipedia.org/wiki/Image:Wilhelm_Carl_Ritter_von_Haidinger_%28edit%29.jpg

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SOLO

Fizeau Experiment

Armand Hyppolite Louise Fizeau used, in 1849,an apparatus consisted of a rotating toothed wheel and a mirror at a distance of 8833 m.

Speed of Light

A toothed wheel rotated at the focal point of the lens L2 in Figure above. A pulse oflight passes at the opening between teeth passes through L2 and is returned by thespherical mirror back to the toothed wheel. The rotation speed of the wheel is adjustedsuch that the light can either pass or be obstructed by a tooth (it was 25 rev/sec). The apparatus is not very accurate since the received light intensity must be minimized to obtain the light velocity.

Fizeau obtained 315,300 km/sec for the light velocity.

( ) ( ) st 000,18/125/1720/1 =⋅= skmt

dv /000,311

000,18/1

633,822 =⋅==

1849


a, 06/10/2006

1. Jenkins & White, "Fundamentals of Optics", p.7

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InterferenceSOLO

Jamin’s Interferometer

1856

J. Jamin, C.R. Acad. Sci. Paris, 42, p.482, 1856

Jules Célestin Jamin1818 - 1886

S

2T

1T 1C

2CD

D

1C 2C

E

1G

2G

1

2

Jamin's Interferometer

In the Jamin’s Interferometer a monochromaticlight from a broad source S is broken into two parallel beams by two parallel faces of a tick plate of glass G1. These two rays pass through to another identical plate of glass G2 to recombine after reflection, forming interference fringes. If the plates are parallel the paths are identical.

To measure the refractive index of a gas asfunction of presure and temperature, two identicalempty tubes T1 and T2 are placed in the twoparallel beams. The gas is slowly introduced in one of the tube.The number of fringes Δm is counted when the gas reaches the desired pressure and temperature. The compensating plates C1 and C2, of equalthickness are rotated by the single knob D. Onepath length is shorted the other lengthened tocompensate the difference between the paths.


http://de.wikipedia.org/wiki/Bild:Jules_Celestin_Jamin.jpg

24

SOLO

Fizeau Experiment in Moving Media

In 1859 Fizeau described an experiment performed to determine the speed of light in moving medium.

Speed of Light

The light of source S placed at the focus of lens L1 passes trough a tube with flowing water, focused by lens L2 to the mirror that reflects it to a second tube in which the water flows with the same velocity u in opposite direction. The returning light is diverted by thehalf silvered mirror an interferes with the light from the source.

Fizeau measured the shift between the fringes obtained when is no water flow and those obtained when the water flows. He fitted the following formula for the speed of light v in moving media to the speed of light v0 in stationary media, with index of refraction n:

−+=

20

11

nuvv

1859

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InterferenceSOLO

Fizeau Fringes (1862)

Spacer

Beam-splitter

ExtendedSources

ViewingScreenDielectric

film

Fizeaufringes

xα

1n

fn

2n

Reference

Test Surface

Test Surface

Reference

Beam Slitter

Eye

Source

θx

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InterferenceSOLO

Fizeau Fringes (1862)

Spacer

Beam-splitter

ExtendedSources

ViewingScreenDielectric

film

Fizeaufringes

xα

1n

fn

2n

Reference

Test Surface

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InterferenceSOLO

Fizeau Interferometer

1862

Fizeau, C.R. Acad. Sci. Paris, 66, p.429, 1862 J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu


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SOLO

Foucault Experiment Foucault working with Fizeau improved the apparatus by replacing the toothed wheel with a rotating mirror. In 1850 Foucault used theimproved apparatus to measure the speed of light in air and in water.In 1862 he used an improved version to give an accurate measurementof speed of light in air.

Speed of Light

The solar light passes trough half silvered mirror, through lens L. It is reflected by the Rotating mirror to a Spherical Mirror at a distance of d = 20 m, back to rotating mirror, through L to half silvered mirror to the Display. When the rotting mirror is stationary theray reaches the point A on the Display. When the rotting mirror is rotating at angular rate

ω, the ray to the spherical mirror will change direction by an angle α = ωτ (τ = d/c)

and the displayed ray by an angle 2 α, reaching point A’ on the Display.

1850 - 1862

29

SOLO

Foucault Experiment (continue – 1)

Speed of Light

http://micro.magnet.fsu.edu/primer/lightandcolor/speedoflight.html

The Spherical Mirror was at a distance d = 20 m from the Rotating Mirror, that rotated at 1,000 revolutions per second, given a displacement of AA’ of 1 mm.

The speed of light obtained by Foucault was 298,000 km/sec.

In 1850 Foucault completed the measurement of speed of light in water.

Newton’s corpuscular light model required that the speed of light in optically dense media be greater than in air, whereas the wave theory, as initiated by Huyghens, correctly predicted that the speed of light must be smaller in optically dense media,and this was verified by Foucault experiments.


30

SOLO

Michelson Interferometer – Interference Fringes

Interference 1882 Nobel Prize 1907

José Antonio Diaz Navashttp://www.ugr.es/~jadiaz/

+

+++= π

λπ

f

yxdIIIII

22

2121 2cos2

cos2

I – intensity of the interference fringes

I1, I2 – intensity of the intensities of the two beamsλ – wavelengthd – path length difference between the two interferometers arms

x,y – coordinates of the focal plane of a lens of focal length f

“Interference Phenomena in a new form of Refractometer”,American J. of Science (3), 23, (1882), pp.392-400 andPhilos. Mag. (5) 13 (1892), pp.236-242

31

SOLO

Michelson Interferometer – Broad Source

Interference 1882 Nobel Prize 1907

José Antonio Diaz Navashttp://www.ugr.es/~jadiaz/

[ ]

+

+

+∆

−++= πλπ

υπ

f

yxd

Logc

yxf

d

IIIII22

22

2121 2cos2

cos2

1cos

exp2

I – intensity of the interference fringes

I1, I2 – intensity of the intensities of the two beams

λ, c – wavelength, speed of lightd – path length difference between the two interferometers armsx,y – coordinates of the focal plane of a lens of focal length f

The intensity of the interference fringes for a Michelson interferometer having a source emitting with a Gaussian profile having a bandwidth of Δν is given by

32

SPECIAL RELATIVITY

Michelson and Morley Experiment

Albert AbrahamMichelson

1852 - 1931

Edward W. Morley

Mikelson and Morley attempted to detect the motion of earth through the aether by comparing the speed of light in the earth direction movement in the orbit around the sun with the perpendicular direction of this movement.

They failed to find any differences, a result consistent with a fixed speed of lightand Maxwell’s Equations but inconsistent with Galilean Relativity.

SOLO

1887


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InterferometersSOLO

Ernst Mach and Ludwig Zehnder separately described what has become the Mach-Zehnder Interferometer.

1891/92

Ludwig Louis AlbertZehnder

1854-1949

Ernst Waldfried Joseph WenzelMach

1838 - 1916

Ernst Mach. “Modifikation und Anwendung des Jamin Interferenz-Refraktometers”.Anz. Acad. Wiss. Wien math. Naturwiss. Klasse 28, p.223-224, 1981

Ludwig Zehnder, “Ein neuer Interferenzrefractor”, Z.Instrumentenkd. 11, p.275-285,1981

Mach-Zehnder Interferometer


http://upload.wikimedia.org/wikipedia/en/9/9c/Mach-zehnder.PNG

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InterferenceSOLO

Sirks-Pringsheim Interferometer

1893/1898

J. A. Sirks, Hd. Ned. Nat. en Geneesk. Congr., Groningen, p. 92, 1893E. Pringsheim, Verh. Phys. Ges., Berlin, p. 152, 1898

A variant of the Jamin interferometer, using plates which are slightly wedge shaped instead of plane parallel, was developed by Sirks and later by Pringsheim for the refractive index of small objects.

Corresponding to an incident ray SA in a principal section of the wedges, the two rays SABCG, SADEF which leave the second plate intersect virtually at a point P behind the second plate. With a quasi-monochromatic source there fringes apparently localized in the vicinity of P, which can be observed with the microscope M. The fringes at P run at right angles to the plane defined by the two emergent rays, i.e. parallel to the wedge apexes. The Object O to be examined is placed between the plates in the path of the ray CG. The image P’ of P in the front of the surface of the second plate also lies on CG, at a position which depends on the inclination of the plates. Return to Table of Content

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InterferenceSOLO

Rayleigh’s Interferometer

1896

Lord Rayleigh, Proc. Roy. Soc.,59, p. 198, 1896

Nobel Prize 1904

Rayleigh’s Interferometer is used to measure the refractiveindex of a gas. His refractometer is based on Young’s doubleslit interferometer. The two coherent rays passing through the slits S1 and S2, from the single source S passed through the tubes T1 and T2 filed with the gas. When the pressure of the gas is changed in on of the tube a difference in the refraction index occurs, the optical paths of the two rays change and the fringe system, viewed at the eyepiece E located at the focus of the second lens, changes.

A count of the fringes as they moved provides a measurement of optical path change, therefore of the refractive index.

The compensating plates C1 and C2, of equalthickness are rotated by the single knob D. Onepath length is shorted the other lengthened tocompensate the difference between the paths.

S

1S

2S2T

1T

f

1C

2CD

D

1C 2C

E

Rayleigh's Interferometer


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InterferenceSOLO

Fabry – Perot Interferometer 1899

http://en.wikipedia.org/wiki/Fabry-Perot_interferometer

Marie P. A.C. Fabry and Jean B.A. Perot (France) developed the Fabry – Perot Interferometer

Jean-Baptiste Alfred Perot

1863 – 1925

Marie P. A.C. Fabry and Jean B.A. Perot, “Théory et applications d’une nouvelle méthode de spectroscopie interférentielle”, Ann. Chim. Phys. (7), 16, p.115-144, 1899

http://www.daviddarling.info/encyclopedia/F/Fabry.html

Marie Paul August Charles Fabry

1867 – 1945

http://www.patrimoine.polytechnique.fr/collectionhomme/portrait/fabrybig.jpg

This interferometer makes use of multiple reflections between two closely spaced partially silvered surfaces. Part of the light is transmitted each time the light reaches the second surface, resulting in multiple offset beams which can interfere with each other. The large number of interfering rays produces an interferometer with extremely high resolution, somewhat like the multiple slits of a diffraction grating increase its resolution.

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/fabry.html


http://en.wikipedia.org/wiki/Fabry-Perot

3737

Optics HistorySOLO

Sagnac, G. 1913. Comptes Rendus, 157:708 & 1410

1913Sagnac Effect

In 1913, Georges Sagnac showed that if a beam of light is split and sent in two opposite directions around a closed path on a revolving platform, and then the beams are recombined, they will exhibit interference effects. From this result Sagnac concluded that light propagates at a speed independent of the speed of the source. The effect had been observed earlier (by Harress in 1911), but Sagnac was the first to correctly identify the cause.

The Sagnac effect (in vacuum) is consistent with stationary ether theories (such as the Lorentz ether theory) as well as with Einstein's theory of relativity. It is generally taken to be inconsistent with emission theories of light, according to which the speed of light depends on the speed of the source.

Retrieved from "http://en.wikipedia.org/wiki/Georges_Sagnac"

George Sagnac 1869-1926


http://upload.wikimedia.org/wikipedia/en/0/03/Sagnac_effect256x128.gif

38

InterferenceSOLO


1916

T. Twyman and A. Green, British Patent No. 103832, 1916

J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu


39

SOLO

Michelson’s Experiments

Michelson performed a series of experiments to determine the speed of light using a rotating mirror situated at Mount Wilson and a fixed mirror on Mount San Antonio, at a distance of 22 miles (35 km). He obtained an average value of 299,796 km/sec.

Speed of Light 1926


40

InterferenceSOLO

Dyson’s Interferometer

1950

J. Dyson, Proc. Roy. Soc., A., 204, p.170, 1950

Dyson’s Interferometer is a Polarization Interferometer. Light from a source L.S. polarized circularly or linearly at 45º to the plate in Figure is incident on a Wollason Prism WP, which divide it into the reference beam (solid line) polarized perpendicularly to the plane of the Figure. Both beams propagate through a lens L, are reflected from surfaces M3 and M2 , respectively, and propagate again through the lens L.Subsequently, they are transmitted through the Quarter-Wave Plate QWP.

Upon propagating twice through the Quarter-Wave Plate QWP, the Measurement Beam becomes polarized perpendicularly to the plane of the Figure, while the Reference Beam becomes polarized in the plane of the Figure. Both beams are transmitted through the lens L, reflected from surfaces M2 and M3’ respectively. Subsequently they return, through the lens L, to the Wollaston Prism WP, which combine them into one beam whose State of Polarization is a function of the Phase Difference introduced by the measured displacement Δx. Detection Setup DS produces an electrical signal corresponding to the State of Polarization of the Beam, from which the measured displacement Δx can be obtained.


41

InterferenceSOLO

Hanbury-Brown and Twiss Interferometer

1956

Robert Hanbury-Brown and Richard Q. Twiss published “A test of a new typeof stellar interferometer on Sirius”, Nature, vol. 178, pp.1046, 1956

Richard Q. Twiss1920 - 2005

Robert Hanbury-Brown1916 - 2002

the Hanbury Brown and Twiss (HBT) effect is any of a variety of correlation and anti-correlation effects in the intensities received by two detectors from a beam of particles. HBT effects can generally be attributed to the dual wave-particle nature of the beam, and the results of a given experiment depend on whether the beam is composed of fermions or bosons. Devices which use the effect are commonly called intensity interferometers and were originally used in astronomy, although they are also heavily used in the field of quantum optics.


42

InterferenceSOLO

Gires-Tournois Etalon

1964

F. Gires, and P. Tournois (1964). "Interferometre utilisable pour la compression d'impulsions lumineuses modulees en frequence". C. R. Acad. Sci. Paris 258: 6112–6115. (An interferometer useful for pulse compression of a frequency modulated light pulse.)

A Gires-Turnois interferometer is an optic standing-wave cavity designed to create chromatic dispersion.The front mirror is partially reflective, while the backmirror has a high reflectivity. If no losses occur in thecavity, the power reflectivity is unity at all wavelength,but the phase of the reflected light is frequency-dependentdue to the cavity effect, causing group delay dispersion (GDD).


43

InterferenceSOLO

Interference of Two Monochromatic Waves Given two waves ( ω = constant ):

( ) ( ) ( )[ ]{ } ( ){ }tUtiAtAtu 111111 ReexpRecos =+=+= φωφω

where the corresponding phasors, are defined as:

( ) ( )[ ]111 exp: φω += tiAtU

The two waves interfere to give:

( ) ( ) ( ) ( ) ( )( ) ( ){ } ( )φω

φωφω+=+=

+++=+=

tAtUtU

tAtAtututu

cosRe

coscos

21

221121


( ) ( )[ ]222 exp: φω += tiAtU

1U

2U

21 UUU +=

1φ2φ φ

( )1221

2

2

2

1

212211

cos2

2

φφ −⋅⋅++=

⋅⋅+⋅+⋅== ∗∗∗

AAAA

UUUUUUUA

2U

( )

++=

−++=

−

2211

22111

2121

2

2

2

1

coscos

sinsintan

cos2

φφφφφ

φφ

AA

AA

AAAAA

The Phasor summationis identical toVector summation

44

InterferenceSOLO

Interference of Monochromatic Waves Given two electromagnetic monochromatic ( ω = constant ) waves:

( ) ( ) ( ) ( ) ( )[ ]{ } ( ){ }trErktirErktrEtrE ,ReexpRecos, 1111110111110111

=+⋅−=+⋅−= φωφω

( ) ( ) ( ) ( ) ( )[ ]{ } ( ){ }trErktirErktrEtrE ,ReexpRecos, 2222220222220222

=+⋅−=+⋅−= φωφω

where the corresponding phasors, are defined as:

( ) ( ) ( )[ ]11110111 exp:, φω +⋅−= rktirEtrE

( ) ( ) ( )[ ]22220222 exp:, φω +⋅−= rktirEtrE

1S

2S

P

1r

2r

2211 12

:&12

: rkrkλπ

λπ ==

At the point P the two waves interfere to give:

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ){ }trEtrE

rktrErktrEtrEtrEtrE

,,Re

coscos,,,

2211

2222021111012211

+=

+⋅−++⋅−=+= φωφω

The Irradiance at the point P is given by:

( ) ( ) ( ) ( )trHtrHtrEtrEI ,,,, ∗∗ ⋅=⋅= µε

45

InterferenceSOLO

Interference of Monochromatic Waves

1S

2S

P

1r

2r

The Irradiance at the point P is given by:

( ) ( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )trEtrEtrEtrEtrEtrEtrEtrE

trEtrEtrEtrEtrEtrEI

,,,,,,,,

,,,,,,

1122221122221111

22112211

∗∗∗∗

∗∗∗

⋅+⋅+⋅+⋅=

+⋅+=⋅=

εεεεεε

( ) ( ) ( ) ( )10110111111 ,, rErEtrEtrEI

⋅=⋅= ∗ εε

( ) ( ) ( ) ( )20220222222 ,, rErEtrEtrEI

⋅=⋅= ∗ εε

( ) ( ) ( ) ( )( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( )[ ] ( )[ ]{ }

( ) ( ) ( ) ( )21112221211122202101

211122211122202101

111101222202

222202111101

1122221112

cos2cos2

expexp

expexp

expexp

,,,,

φφφφε

φφφφε

φωφωε

φωφωε

εε

−+⋅−⋅=−+⋅−⋅⋅=

−+⋅−⋅−+−+⋅−⋅⋅=

+⋅−−⋅+⋅−+

+⋅−−⋅+⋅−=

⋅+⋅= ∗∗

rkrkIIrkrkrErE

rkrkirkrkirErE

rktirErktirE

rktirErktirE

trEtrEtrEtrEI

( )21112221211221 cos2 φφ −+⋅−⋅++=++= rkrkIIIIIIII

46

InterferenceSOLO

Interference of Monochromatic Waves 1S

2S

P

1r

2r

The maximum Irradiance at the point P is given by:

,2,1,0&22 2111222121max ±±==−+⋅−⋅←++= mmrkrkIIIII πφφ

The minimum Irradiance at the point P is given by:

( )

,2,1,0&122 2111222121min ±±=+=−+⋅−⋅←−+= mmrkrkIIIII πφφ

2211 12

:&12

: rkrkλπ

λπ ==

Since

,2,1,0&2

2 21122121max ±±==−+−←++= mmrrIIIII λλ

πφφ

( ) ,2,1,0&2

122

2 21122121min ±±=+=−+−←−+= mmrrIIIII

λλπφφ

The Visibility of the fringes is defined as:21

21

minmax

minmax2

:II

II

II

IIV

+=

+−

=


47

InterferenceSOLO

Billet’s Split Lens

Meslin’s Experiment

Two Basic Classes of Interferometers

• Division of Wavefront (portion of the primary wavefront are used either directly as sources to emit secondary waves or in conjunction with opticaldevices to product virtual sources of secondary waves.

The primary and secondary waves recombine and interfere)

• Division of Amplitude (the primary wave itself is divided into two waves,

which travel different paths before recombining and interfering)

Beamsplitter

Diffraction

Young’s Experiment

Fresnel’s Double Mirror

Fresnel’s Bi-prism

Lloyd’’s Mirror (1834) mirror

48

Two Basic Classes of Interferometers InterferenceSOLO


49

Wavefront-Splitting Interferometer SOLO

Young’s Experiment

1r

2r

s

a

y

2S

1S

P

OS 'O

aΣoΣ

Young passed sun light through a pinhole,which become the primare source, obtaineda spatially coherent beam through twoidentically illuminated apertures. The twoapertures acted as two coherent sourcesproducing a system of alternating bright anddark bands of interference fringes. Given a point P on the screen at distancesr1 and r2 from apertures S1 and S2, respectively. We have

The path difference is:

2

2

2

22

2

2

2

11 22zy

asPSrzy

asPSr +

++==+

−+==

( ) ( ) ( )s

ay

s

z

s

ay

s

z

s

ays

zya

szya

srr

ays

=

+−−−

+++≈

+

−+−+

++=−

+>>

2

2

2

2

2

2

2

22/

2

2

22

2

2

12

2/

2

11

2/

2

11

22

http://homepage.univie.ac.at/Franz.Embacher/KinderUni2005/waves.gif

2S

1S

y

z

50

SOLO

Young’s Experiment (continue – 1)

s

ayrr

sa<<

≈− 12

The bright fringes are obtained when:,2,1,012 ±±==− mmrr λ

,2,1,0 ±±== mms

ay λ

The distance between two consecutive bright fringes is:

( ) λλλa

s

a

sm

a

smyyy mm =−+≈−=∆ + 11

The dark fringes are obtained when:

,2,1,0212 ±±=+=− mmrrλλ

( ) ,2,1,02

12 ±±=+= mms

ay

λ

λ - wavelength

The Intensity at point P is:

( ) ( )[ ]{ }

=−+=−+⋅−⋅++=

=

−=−

==

= s

yaIrrkIrkrkIIIII

k

syarr

III

λπφφ

λπ

φφ

2

0

/2

/1202111222121 cos4cos12cos212

021

21

1r

2r

s

a

y

2S

1S

P

OS 'O

aΣoΣ

http://homepage.univie.ac.at/Franz.Embacher/KinderUni2005/waves.gif

Wavefront-Splitting Interferometer

Classes of InterferometersReturn to Table of Content

51

SOLO

http://info.uibk.ac.at//c/c7/c704/museum/en/details/optics/fresnel.html

University of Innsbruck

Fresnel’s Double Mirror consists oftwo planar mirrors inclined to eachother at a very small angle δ.



1788-1827

Fresnel’s Double Mirror (1819*)

52

SOLO

Fresnel’s Double Mirror (continue – 1)

Fresnel’s Double Mirror consists oftwo planar mirrors inclined to eachother at a very small angle δ. The slit S image of the first mirroris S1 and of the second mirror is S2.The points S, S1 and S2 determinea plane normal to both planar mirrorsthat intersects them at a point C (onthe intersection line of the two mirrors)

We have: RCSCSSC === 21

δ=∠ 21 SSS

Since is normal to the first mirrorand is normal to second mirror, we have:

1SS

2SS

Also: δ⋅=∠⋅=∠ 22 2121 SSSSCS

a1S

2S

S

R

R

R

Screen

Schield

Mirror 2

C

δ

δδ

δ

sMirror 1

SC

IC

We will arrange a planar screen perpendicular to the normal from point C to line, , that also bisects the angle .aSS =21

δ221 =∠ SCSCC I

– is the distance between line and the screen.21SSSI CCs = A shield is introduced to prevent the waveform to travel straight from slit S to screen.


53

SOLO


From the slit S a cylindrical waveformis reflected by one side of the mirror atpoint A and reaches the screen at pointP, while an other cylindrical waveformis reflected by the other side of the mirrorat point B and interferes with the first atthe point P on the screen.

Because of the reflection:

BSSBASSA 21 & == Therefore we have:

111 rPSAPASAPSA ==+=+

222 rPSBPBSBPSB ==+=+

a1S

2S

S

AB

P

R

R

R

Screen

Schield

Mirror 2

C

δ

δδ

δ

s

2r1r

Mirror 1

y

SC

IC

where: – is the distance between line and the screen.21SSSI CCs =

PCySSa S== &21

2

2

2

22

2

2

2

11 22zy

asPSrzy

asPSr +

++==+

−+==


54

SP

R

Screen

Schield

Mirror 2

C

δ

sMirror 1

Slit

SCy

z

SOLO


We have:


The bright fringes are obtained when:

( ) λλλa

s

a

sm

a

smyyy mm =−+≈−=∆ + 11

a1S

2S

S

AB

P

R

R

R

Screen

Schield

Mirror 2

C

δ

δδ

δ

s

2r1r

Mirror 1

y

SC

IC

S

AB

P

R

Screen

Schield

Mirror 2

C

δ

sMirror 1

Slit

SCy

z

S

AB

P

R

Screen

Schield

Mirror 2

C

δ

sMirror 1

Slit

ySC

z

P

2

2

2

22

2

2

2

11 22zy

asPSrzy

asPSr +

++==+

−+==

( ) ( ) ( )s

ay

s

z

s

ay

s

z

s

ays

zya

szya

srr

ays

=

+−−−

+++≈

+

−+−+

++=−

+>>

2

2

2

2

2

2

2

22/

2

2

22

2

2

12

2/

2

11

2/

2

11

22


,2,1,0&2

2112 ±±==−+− mmrr λλ

πφφ

Since the distance between two consecutive bright fringes is:

21 φφ =

Classes of Interferometers


55

SOLO

Fresnel’s Double Prism

The Fresnel’s Double Prism or Bi-prism consists of two thin prisms joined at their bases. A singlr cylindrical wave emerge froma slit. The top part of the wave-front is Refracted downward, and the lower segment is refracted upward. In the region of superposition interference occurs.

Screen

Bi-prism

Slit

y

z

δ

s

a

2S

1S

OS 'O

aΣoΣ

1<<α

iθ

d

iθ - incident angle

δ - dispersion angle

α - prism angle

From the Figure we can see that twovirtual sources S1 and S2 exists. Let abe the distance between them.From the Figure

( ) δδθθθ

δddd

a i

ii

1

1sintan

2

<<

<<≈−−=

where

θi – ray incident angleδ – ray dispersion (deviation) angled – distance slit to bi-prism vertexα – prism angle

( )[ ][ ] ( ) ααθαθ

αθαθαθδα

θ

α

θ1sin

sincossinsinsin1

1

11

1

2/1221

−≈−−+≈

−−−+=<<

<<

−<<

<<

−

nn

nn

ii

iii

ii

See δ development


56

SOLO

Dispersive Prisms( ) ( )2211 itti θθθθδ −+−=

21 it θθα +=

αθθδ −+= 21 ti

202 sinsin ti nn θθ =Snell’s Law

10 ≈n

( ) ( )[ ]1

1

2

1

2 sinsinsinsin tit nn θαθθ −== −−

( )[ ] ( )[ ]11

21

11

1

2 sincossin1sinsinsincoscossinsin ttttt nn θαθαθαθαθ −−=−= −−

Snell’s Law 110 sinsin ti nn θθ =11 sin

1sin it n

θθ =10 ≈n

( )[ ]12/1

1

221

2 sincossinsinsin iit n θαθαθ −−= −

( )[ ] αθαθαθδ −−−+= −1

2/1

1

221

1 sincossinsinsin iii n

The ray deviation angle is

Optics - Prisms

57

SOLO

Fresnel’s Double Prism (continue – 1)

From the Figure we found that the distance abetween virtual sources S1 and S2 is:

( ) δδθθθ

δddd

a i

ii

1

1sintan

2

<<

<<≈−−=

( )[ ][ ] ( ) ααθαθ

αθαθαθδα

θ

α

θ1sin

sincossinsinsin1

1

11

1

2/1221

−≈−−+≈

−−−+=<<

<<

−<<

<<

−

nn

nn

ii

iii

ii

See δ development

( ) α12 −≈ nda

s

a

y

2S

1S P

OS 'O

aΣoΣ

1<<α

α - prism angle

1r

2r

Screen

Bi-prism

Slit

y

z

Consider two rays starting from the slit S thatpass the bi-prism and interfere on the screen at P. We can assume that they are strait lines starting at the virtual source S1 and S2, andhaving optical paths r1 and r2, respectively.

2

2

2

22

2

2

2

11 22zy

asPSrzy

asPSr +

++==+

−+==


δ

s

a

2S

1S

OS 'O

aΣoΣ

1<<α

iθ

d

iθ - incident angle

δ - dispersion angle

α - prism angle

58

SOLO

Fresnel’s Double Prism (continue – 2)

( ) α12 −≈ nda

s

a

y

2S

1S P

OS 'O

aΣoΣ

1<<α

α - prism angle

1r

2r

d

Screen

Bi-prism

Slit

y

z

2

2

2

22

2

2

2

11 22zy

asPSrzy

asPSr +

++==+

−+==



( ) λλλa

s

a

sm

a

smyyy mm =−+≈−=∆ + 11

( ) ( ) ( )s

ay

s

z

s

ay

s

z

s

ays

zya

szya

srr

ays

=

+−−−

+++≈

+

−+−+

++=−

+>>

2

2

2

2

2

2

2

22/

2

2

22

2

2

12

2/

2

11

2/

2

11

22

We have:


,2,1,0&2

2112 ±±==−+− mmrr λλ

πφφ

Since the distance between two consecutive bright fringes is:

21 φφ =



59

SOLO

Lloyd’s Mirror Interferometer

The Lloyd’s planar mirror is perpendicularto the planar screen. A cylindrical waveformfrom the slit S is reflected by the mirror andinterferes at the screen with the portion of thewave that proceeds directly to the screen. Screen

Plane Mirror

Slit

y

z

From the Figure we can see that avirtual source S1, that is symmetric relativeto mirror plane exists. The slit, parallel tomirror plane, is at the same distance, a/2,from the mirror plane as it’s virtual image.


sa

y

1S

P

O

S

oΣ

1r

2r2/a

2/a

Planar Mirror

Screen

Screen

Plane Mirror

Slit

y

z

Consider two rays starting from the slit S, oneproceeding directly to the screen and the otherreflected by the mirror and interfere on the screen at P. We can assume that they are strait lines starting at S and at the the virtual source S1, andhaving optical paths r1 and r2, respectively.

2

2

2

12

2

2

2

1 22zy

asPSrzy

asPSr +

++==+

−+==

Humphrey Lloyd1800-1881

60

SOLO

Lloyd’s Mirror Interferometer (continue – 1)


sa

y

1S

P

O

S

oΣ

1r

2r2/a

2/a

Planar Mirror

Screen

Screen

Plane Mirror

Slit

y

z



The distance between two consecutive bright fringes is:

( ) λλλa

s

a

sm

a

smyyy mm =−+≈−=∆ + 11

( ) ( ) ( )s

ay

s

z

s

ay

s

z

s

ays

zya

szya

srr

ays

=

+−−−

+++≈

+

−+−+

++=−

+> >

2

2

2

2

2

2

2

22/

2

2

22

2

2

12

2/

2

11

2/

2

11

22

We have:

2

2

2

12

2

2

2

1 22zy

asPSrzy

asPSr +

++==+

−+==

,2,1,0&2

2112 ±±==−+− mmrr λλ

πφφ



61

Stokes Treatment of Reflection and Refraction SOLO

An other treatment of reflection and refraction was given by Sir George Stokes.

Suppose we have an incident wave of amplitude E0i

reaching the boundary of two media (where n1 = ni and n2 = nt) at an angle θ1. The amplitudes of the reflected and transmitted (refracted) waves are, E0i·r and E0i·t, respectively (see Fig. a). Here r (θ1) and t (θ2) are the reflection and transmission coefficients.

According to Fermat’s Principle the situation where the rays direction is reversed (see Fig. b) is also permissible. Therefore we have two incident rays E0i·r in media with refraction index n1 and E0i·t in media with refraction index n2.E0i·r is reflected, in media with refraction index n1, to obtain a wave with amplitude (E0i·r )·t and refracted, in media with refraction index n2, to obtain a wave with amplitude (E0i·r )·r (see Fig. c).

E0i·t is reflected, in media with refraction index n2, to obtain a wave with amplitude (E0i·t )·r’ and refracted, in media with refraction index n1, to obtain a wave with amplitude (E0i·t )·t’ (see Fig. c).

θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =

62

Stokes Treatment of Reflection and Refraction SOLO

An other treatment of reflection and refraction was given by Sir George Stokes(under the assumption that is not absorption of energy at the boundary of the two media).

To have Fig. c identical to Fig. b the following conditions must be satisfied:

( ) ( ) ( ) ( ) iii ErrEttE 0110120 ' =+ θθθθ( ) ( ) ( ) ( ) 0' 220210 =+ θθθθ rtEtrE ii

Hence:

( ) ( ) ( ) ( )( ) ( )12

1112

'

1'

θθθθθθ

rr

rrtt

−==+

Stokes relations

θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =

Let check that Fresnel Equation do satisfy Stokes relations

( )2211

112 coscos

cos221

θθθθ

µµ

nn

nt

+==

⊥

2112

11|| coscos

cos221

θθθµµ

nn

nt

+==

( )2211

22111 coscos

coscos21

θθθθθ

µµ

nn

nnr

+−

==

⊥

( )2112

21121|| coscos

coscos21

θθθθθ

µµ

nn

nnr

+−=

=

Parallel InterfacesReturn to Table of Content

63

SOLO

Optical Reflected Path Length Difference: Parallel Interfaces Two-Beam Interference: Parallel Interfaces

We have a point source and a dielectric slabthat performs a double reflection giving two coherent rays (1) and (2). Using a lens the tworays interfere at lens focus.

'D1θ

1θ

1θ 2θ2θ

d

C

B

D

1n

2n1n

Pointsource

Image

1

2

Dielectricslab

We consider a dielectric slab that has low reflectivity at each interface: r,r’<<1

Assume an incident ray that at point B is( ) ( )tiABEi ωexp=

For the reflected ray (1) we have at point D

( )

−=

0

122

'2exp'

λπω BDn

tiADE

For the reflected ray (2) we have at point D’. DD’ is normal two ray (2) so that both raystravel the same optical paths until interference.

( ) ( )

+−=0

211

2exp

λπω CDBCn

tiADE

Amplitude Split Interferometers

64

SOLO

Optical Reflected Path Length Difference: Parallel Interfaces (continue – 1)

Two-Beam Interference: Parallel Interfaces

To compute the amplitudes A1 and A2 we will use :

'D1θ

1θ

1θ 2θ2θ

d

C

B

D

1n

2n1n

Pointsource

Image

1

2

Dielectricslab

2θ

2θ

1θ

( ) ( ) ( )2211 '' θτθθτ rAA =( )12 θrAA =

Using Stokes relations:

where:

( ) ( )11 , θτθr - reflectivity and transitivity at B

( )2' θr - reflectivity at C

( )2' θτ - transitivity at D from slab to air

( ) ( )12' θθ rr −=( ) ( ) ( )

( )11'1

12

21

12 <<

≈−=θ

θθτθτr

r

( ) ( ) ( ) ( )12211 '' θθτθθτ rArAA −==we obtain:

( )12 θrAA =The minus sign shows that is an additional phase delay of π between ray (1) at point D and ray (2) at point D’.


65

SOLO

Optical Reflected Path Length Difference: Parallel Interfaces (continue – 2) Two-Beam Interference: Parallel Interfaces

'D1θ

1θ

1θ 2θ2θ

d

C

B

D

1n

2n1n

Pointsource

Image

1

2

Dielectricslab

( ) ( )

−=

0

112

'2exp'

λπωθ BDn

tirADE

( ) ( ) ( )

++−= π

λπωθ

0

211

2exp

CDBCntirADE

2cos/ θdCDBC ==From the Figure we obtain:

12 sintan2' θθdBD =The phase difference at interference is:

( )[ ] πλπφφ +−+−=− BDnCDBCn 120

21

2

2

22

2

sinsin

12

21121 cos

sinsin

cos

sinsintan

2211

θθθ

θθθθ

θθnnn

nn =

==

( ) πθλ

ππθθλ

ππθθθλ

πφφ +−=+

−−=+

−−=− 2

0

222

2

2

0

121

2

2

0

21 cos4

sin1cos

22sintan

cos2

2 ndndn

nd


66

SOLO

Optical Reflected Path Length Difference: Parallel Interfaces (continue – 3) Two-Beam Interference: Parallel Interfaces

'D1θ

1θ

1θ 2θ2θ

d

C

B

D

1n

2n1n

Pointsource

Image

1

2

Dielectricslab

( ) ( ) ( )212 exp' φωθ += tirADE

( ) ( ) ( )111 exp φωθ += tirADE

πθλ

πφφ +−=− 20

221 cos

4 nd

The Intensity at the interference is:

( )( ){ } ( ){ }

( )2/sin4

cos12cos12

cos2

20

0210

2111222121

021

122

δ

πδφφ

φφ

I

II

rkrkIIIIIIII

rkrk

=

+−+=−+=

−+⋅−⋅++===

⋅=⋅

where

20

2 cos4

: θλ

πδ nd=

( )122021 ~ θrAIII ==



67

SOLO

Optical Transmitted Path Length Difference: Parallel InterfacesTwo-Beam Interference: Parallel Interfaces


( ) ( ) ( )

+−=0

1212

'2exp'

λπωθ BDnABn

tirADE

( ) ( ) ( )

+++−= π

λπωθ

0

211

2exp

CDBCABntirADE

2cos/ θdCDBCAB ===From the Figure we obtain:

12 sintan2' θθdBD =The phase difference at interference is:

( )[ ] πλπφφ +−+−=− '2

120

21 BDnCDBCn

2

22

2

sinsin

12

21121 cos

sinsin

cos

sinsintan

2211

θθθ

θθθθ

θθnnn

nn =

==

( ) πθλ

ππθθλ

ππθθθλ

πφφ +−=+

−−=+

−−=− 2

0

222

2

2

0

121

2

2

0

21 cos4

sin1cos

22sintan

cos2

2 ndndn

nd

( ) ( ) ( ) ( )010

210 exp

2exp δωθ

λπωθ −=

−= tirA

ABntirABE

20

2

20

20

cos4

:

cos

2:

θλ

πδ

θλπδ

nd

nd

=

=


68

InterferenceSOLO

Haidinger Fringes1846


1795 - 1871

Lens

Beam-splitter

ExtendedSources

ViewingScreen

Dielectricfilm

Blackbackground

Circularfringes



69

SOLO

Haidinger Interference Fringes Two-Beam Interference: Parallel Interfaces

We have:

1θ

1θ

2θ

2θ

d

1n2n

1n

Extendedsource

Focalplane

1P2P

1θ

Dielectricslab

Beamsplitter

Lens

θ

f

x



1795 - 1871




70

InterferenceSOLO

Interference of Many Monochromatic Waves Given two waves ( ω = constant ):


The N waves interfere to give:( ) ( ) ( ) ( )

( ) ( ) ( ){ } ( )φω +=+++=

+++=

tAtUtUtU

tutututu

N

N

cosRe 21

21


1U

NUUUU +++= 21

1φ

2φφ

2U

NU

Nφ

The Phasor summation is identical to Vector summation

( ) ( ) ( )[ ]{ } ( ){ }tUtiAtAtu NNNNNN ReexpRecos =+=+= φωφω

71

InterferenceSOLO

Multiple Beam Interference from a Parallel Film We have a point source and a dielectric slab that performs a multiple reflection and transmission.

72

InterferenceSOLO

Multiple Beam Interference from a Parallel Film

( )

( )

( )[ ]

δω

δω

δω

ω

10

32

20

33

02

01

''

''

''

−−−

−

−

=

=

=

=

NtiNrN

tir

tir

tir

eEtrtE

eEtrtE

eEtrtE

eErE

We have:

We have a point source and a dielectric slab that performs a multiple reflection and transmission.

( )

( )

( ) ( )[ ]

δωδ

δωδ

δωδ

ωδ

10

12

20

43

02

2

01

0

0

0

0

''

''

''

'

−−−−

−−

−−

−

=

=

=

=

NtiiNrN

tiit

tiit

tiit

eeEtrtE

eeEtrtE

eeEtrtE

eeEttE 20

2

20

20

cos4

:

cos

2:

θλ

πδ

θλπδ

nd

nd

=

=

73

InterferenceSOLO

Multiple Beam Interference from a Parallel FilmUsing lens the multi-rays interfere at lens focus.

( )[ ] tiNiNii

rNrrr

eEetrtetrtetrtr

EEEEωδδδ

013223

21

''''''

+++++=

++++=−−−−−

( )ti

i

NiNi eE

er

eretrtr ω

δ

δδ

02

132

'1

'1''

−

−+= −

−−−−

∞→<

Nand

rIf 1' tii

i

r eEer

etrtrE ω

δ

δ

02'1

''

−

+= −

−

In the case of zero absorption, no energy being taken out of the waves, using Stokes relations

21'&' rttrr −=−=

( ) tii

i

r eEer

erE ω

δ

δ

021

1

−

−= −

−

∞→<

Nand

rIf 1

74

InterferenceSOLO

Multiple Beam Interference from a Parallel FilmUsing lens the multi-rays interfere at lens focus.

( ) ( )[ ] ( )00

112242

21

''''1 δωδδδ −−−−−− +++++=

++++=tiNiNii

tNttt

eEttererer

EEEE

( )002

2

''1

'1 δωδ

δ−

−

−

−−= ti

i

NiN

eEtter

er

∞→<

Nand

rIf 1' ( )002'1

' δωδ

−−−

= tiit eE

er

ttE

In the case of zero absorption, no energy being taken out of the waves, using Stokes relations

21'&' rttrr −=−=

( )002

2

1

1 δωδ

−−−

−= tiit eE

er

rE

20

2

20

20

cos4

:

cos

2:

θλ

πδ

θλπδ

nd

nd

=

=

75

InterferenceSOLO


∞→<

Nand

rIf 1( ) ti

i

i

r eEer

erE ω

δ

δ

021

1

−

−= −

−

( )002

2

1

1 δωδ

−−−

−= tiit eE

er

rE

Let compute the Reflected and Transmitted Irradiances:

( ) ( ) ( )( ) 024

2*

0022

*

cos21

cos12

1

1

1

1I

rr

rEE

er

er

er

erEEI

i

i

i

i

rrr δδ

δ

δ

δ

δ

−+−=

−

−−

−=∝ −

−

( )( ) 024

22*

002

2

2

2*

cos21

1

1

1

1

1I

rr

rEE

er

r

er

rEEI

iittt δδδ −+−=

−−

−−=∝ −

Using lens the multi-rays interfere at lens focus we foundthat in the case of zero absorption, no energy being taken out of the waves, using Stokes relations 21'&' rttrr −=−=

0III tr =+

( ) ( )[ ] ( )( )[ ] ( ) 0222

2222/sin21cos

2/sin1/21

2/sin1/22

Irr

rrIr

δδδδ

−+−=

−=

( )

( )[ ] ( ) 0222

2/sin21cos

2/sin1/21

12

Irr

Itδ

δδ

−+=

−=

We see that

76

The transmission of an etalon as a function of wavelength. A high-finesse etalon (red line) shows sharper peaks and lower transmission

minima than a low-finesse etalon (blue).

InterferenceSOLO



http://en.wikipedia.org/wiki/File:Etalon-2.png

77

Gas RefrectometerSOLO

S

1S

2S2T

1T

f

1C

2CD

D

1C 2C

E

Rayleigh's Interferometer

t

To measure the refractive index of a gas we can use any interferometer that splits the source ray in two coherent rays passing through the tubes T1 and T2 filed with the gas.

When the pressure of the gas is changed in on of the tube a difference in the refraction index occurs, the optical paths of the two rays change and the fringe system, viewed at the eyepiece E, changes.

A count of the fringes as they moved provides a measurement of optical path change, therefore of the refractive index.

Jamin, Mack-Zehnder or Reyleigh’s interferometers can be used..

S

2T

1T 1C

2CD

D

1C 2C

E

1G

2G

1

2

Jamin's Interferometer

t

S

1T

E

1M

2G

1

2

Mach-ZehnderInterferometer

2M

3M

4M2T

t

( ) λmtntTpn ag ∆=−1

,

( ) tmTpng /1, λ∆+=

The index of refraction of the gas is given by the Lorenz-Lorentz formula (1890/1)

( )

+−+=

2

1

2

31,

2

2

n

nVNTpng

Reyleigh’s Interferometer

Jamin’sInterferometer


78

Interferometers HistorySOLOReferences

M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986 , Ch. 5, Interference

M. Born, E. Wolf, “Principles of Optics”, Pergamon Press,6th Ed., 1980, Ch. VII,Elements of the Theory of Interference and Interferometers

S.G. Lipson, H. Lipson, “Optical Physics”, Cambridge University Press, 1969, Ch. 7,Fraunhofer Diffraction and Interference

E. Hecht, “Optics”, Addison Wesley, 2002, 4th Ed., Ch. 9, Interference

Françon, M., “Optical Interferometry”, Academic Press, 1966

M.V.Klein,“Optics”, 2nd Ed., John Wiley & Sons, 1970, Ch. 5, Interference

Steel, W.,H., “Interferometry”, Cambridge University Press, 1967

M. Kerker, “Scattering of Light and Other Electromagnetic Radiation”, Academic Press, 1969

J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu/jcwyant/


http://www.optics.arizona.edu/jcwyant

January 5, 2015 79

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

http://upload.wikimedia.org/wikipedia/en/a/ae/RAFAEL_logo.png

80J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu

88

Field and linear interferometers InterferenceSOLO

Double-Slit Interferometer

Fourier-transform Interferometer

Astronomical Interferometer/Michelson Stellar Interferometer

Mireau Interferometer (also known as a Mireau objective) (microscopy)

Multi-Beam Interferometer (microscopy) Watson Interferometer (microscopy)

Linnik Interferometer (microscopy)

Diffraction-Grating Interferometer (white light)

White-light Interferometer (see also Optical coherence tomography) Shear Interferometer (lateral and radial)

http://en.wikipedia.org/wiki/List_of_types_of_interferometers

Michelson InterferometerMach-Zehnder Interferometer

Fabry-Perot Interferometer

Sagnac Interferometer

Gires-Tournois Etalon

http://en.wikipedia.org/wiki/Optical_coherence_tomography

89

Field and linear interferometers InterferenceSOLO

Moire Interferometer (see Moire pattern)

Holographic Interferometer

Near-field InterferometerFringes of Equal Chromatic Order Inteferometer (FECO)

Fresnel Interferometer (e.g. Fresnel biprism, Fresnel mirror or Lloyd's mirror)

Polarization Interferometer (see also Babinet-Soleil compensator) Newton Interferometer (see Newton's rings)

Cyclic Interferometer

Point Diffraction Interferometer

White-light Scatterplate Interferometer (white-light) (microscopy)

Phase-shifting Interferometer

Wedge Interferometer

Schlieren Interferometer (phase-shifting) Talbot Lau Interferometer


Fizeau Interferometer

Rayleigh Interferometer


90

Intensity and nonlinear interferometers

InterferenceSOLO


Intensity Interferometer

Intensity Optical CorrelatorFrequency-Resolved Optical Gating (FROG)

Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER)

Quantum optics interferometers Hong-Ou-Mandel Interferometer (HOM) (see Leonard Mandel)

Interferometers outside optics

Francon Interferometer

Atom InterferometerRamsey InterferometerMini Grail Interferometer

Hanbury-Brown Twiss Interferometer

91

http://www.grahamoptical.com/phase.html

In phase-shifting interferometers, Piezo-electric transducers move the analyzing wavefront with respect to the reference wavefront by a specified phase angle, while a frame-grabber captures a video frame at each position and stores them on the computer. The frame data are then processed by the computer to calculate optical wavefront errors. The software finds aberrations and computes both peak-to-valley (PV) and Root Mean Square (rms) values. The operator has the option to subtract tilt, power, astigmatism, coma, and spherical aberrations from the data. Interactive computer graphics make it easy to interpret the output and numerical data provides quantitative results. The image at the right shows the interferogram as the phase shifter moves the reference surface by 1/4 wave at each step

How Phase Shifting Works

92

http://www.grahamoptical.com/phase.html

How Phase Shifting Works (continue – 1)

However, surfaces aren't always that flat, and the interferogram is not always so simple, The example at the left is a surface which is slightly convcave and somewhat irregular. Attempting to interpret the meaning of this fringe pattern is substantially more difficult than when the fringes are better behaved. That is why phase-shifting interferometers are needed to accurately evaluate surface configuration of any but the simplest surfaces.

The Model 2VP PHASE MITE Interferometer shown below is equipped with Durango Universal Interferometry Software. It is just one of Graham's Phase-Shifting Interferometers. Click on the link for further information on other available Phase-Shifting interferometers manufactured by GRAHAM. Click on the following link for further information on Durango.