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Interferometers history

The document provides a comprehensive overview of the history and development of interferometry, diffraction, and optics, highlighting key experiments and scientists from the 17th to 19th centuries. It details significant contributions from figures like Fizeau, Young, Hooke, Newton, and Fresnel, while elucidating concepts such as light waves, interference, refraction, and diffraction patterns. The text also discusses various experimental setups and their implications for understanding the behavior of light.

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1 INTERFEROMETERS HISTORY SOLO HERMELIN Updated: 6.01.11 4.01.15 http://www.solohermelin.com
2 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 Interferometer Gires-Tournois Etalon
3 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 Experiment Division of Wavefront Optical Reflected Path Length Difference: Parallel Interfaces Amplitude Split Interferometers Stokes Treatment of Reflection and Refraction Optical Transmitted Path Length Difference: Parallel Interfaces Haidinger Interference Fringes Interference of Many Monochromatic Waves Gas Refrectometer References
4 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 in 1665 , 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 the Jesuit college in Bolognia discovered the diffraction of light and gave it the name diffraction, 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 be which is formed by the direct rays. Furter it should not be omitted that the illuminated base IK appears in the middle suffused with pure light, and either extremity its light is colored.” Single Slit Diffraction Double Slit Diffraction 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 compound microscope: on the left the illumination device (an oil lamp), on the 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-front derivation 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 OpticsSOLO 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 the corpuscular theory. This conviction was due to the fact 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.
8 InterferenceSOLO Newton Fringes Colllimator Lens Beam- splitter Point Source Viewing Screen Optical flat Circular fringes Black Surface
9 OpticsSOLO History (continue) In 1801 Thomas Young uses constructive and destructive interference of waves to explain the Newton’s rings. Thomas Young 1773-1829 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 spectrum is composed by a number of dark lines, but the interpretation of this phenomenon was done by Fraunhofer in 1814. Wollaston developed In 1802 the refractometer, an instrument used to measure the refractive index. 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, used in microscopy. 1802-1807
11 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 light passing through this combination with a small telescope eypiece. By this technique he was able to investigate the spectrum bit by bit, color by color.
12 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. Diffraction http://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.”
13 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. Diffraction http://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 Arago 1786-1853 Siméon Denis Poisson 1781-1840 Pierre-Simon Laplace (1749-1827) Joseph Louis Guy-Lussac 1778-1850 Judging Committee of French Academy 1818
14 SOLO Diffraction http://microscopy.fsu.edu/optics/timeline/people/gregory.html http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html Dominique François Jean Arago 1786-1853 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 Poisson 1781-1840 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 two rays polarized at right angles to each other never interface. SOLO History (continue) Dominique François Jean Arago 1786-1853 Augustin Jean Fresnel 1788-1827 Arago relayed to Thomas Young in London the results of the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillations in 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 longitudinal waves transversal waves
16 DiffractionSOLO History of Diffraction Augustin Jean Fresnel 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
17 DiffractionSOLO Augustin Jean Fresnel 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 Q 1x 0x 1y 0y η ξ Fr  Sr  ρ  r  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 source Q e r A tU − = ' ' ω According to Huygens Principle second wavelets will start at the aperture and will add at the image point P. ( ) ( ) ( )( ) ( ) ( )( ) ∫∫ Σ ++− Σ +−− == dre rr A Kdre r U KtU rrktj S sourcerkttjQ P S 2/2/' ',', πωπω θθθθ where: ( )',θθK obliquity or inclination factor ( ) ( )SSS nrnr 11cos&11cos' 11 ⋅=⋅= −− θθ ( ) ( )   === === 0',0 max0',0 πθθ θθ 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:
18 SOLO History In 1821 Joseph Fraunhofer build the first diffraction grating, made up of 260 close parallel wires. Latter he built a diffraction grating 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 monochromatic unpolarized light is normal to one of the crystal faces, the energy will spread out in the plate in a hollow cone, the cone 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 Hamilton in 1832 and confirmed experimentally by Humphrey Lloyd, a year later (Born & Wolf). Because it is no easy to obtain an accurate parallel beam of monochromatic light on obtained two bright circles (Born & Wolf). 1832 William Rowan Hamilton (1805-1855) Humphrey Lloyd (1800-1881)
20 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
21 InterferenceSOLO Haidinger Fringes 1846 Wilhelm Karl, Ritter von Haidinger 1795 - 1871 Lens Beam- splitter Extended Sources Viewing Screen Dielectric film Black background Circular fringes Haidinger Fringes are the type of interference pattern that results with an extended source where partial reflections occur from a plane-parallel dielectric slab. Return to Table of Content
22 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 of light passes at the opening between teeth passes through L2 and is returned by the spherical mirror back to the toothed wheel. The rotation speed of the wheel is adjusted such 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 =⋅= skm t d v /000,311 000,18/1 633,822 = ⋅ == 1849 Return to Table of Content
23 InterferenceSOLO Jamin’s Interferometer 1856 J. Jamin, C.R. Acad. Sci. Paris, 42, p.482, 1856 Jules Célestin Jamin 1818 - 1886 S 2 T 1 T 1C 2C D D 1 C 2C E 1G 2 G 1 2 Jamin's Interferometer In the Jamin’s Interferometer a monochromatic light 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 as function of presure and temperature, two identical empty tubes T1 and T2 are placed in the two parallel 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 equal thickness are rotated by the single knob D. One path length is shorted the other lengthened to compensate the difference between the paths. Return to Table of Content
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 the half 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 1 1 n uvv 1859
25 InterferenceSOLO Fizeau Fringes (1862) Spacer Beam- splitter Extended Sources Viewing ScreenDielectric film Fizeau fringes x α 1n f n 2 n Reference Test Surface Test Surface Reference Beam Slitter Eye Source θ x
26 InterferenceSOLO Fizeau Fringes (1862) Spacer Beam- splitter Extended Sources Viewing ScreenDielectric film Fizeau fringes x α 1n f n 2 n Reference Test Surface
27 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 Return to Table of Content
28 SOLO Foucault Experiment Foucault working with Fizeau improved the apparatus by replacing the toothed wheel with a rotating mirror. In 1850 Foucault used the improved apparatus to measure the speed of light in air and in water. In 1862 he used an improved version to give an accurate measurement of 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 the ray 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. Return to Table of Content
30 SOLO Michelson Interferometer – Interference Fringes Interference 1882 Nobel Prize 1907 José Antonio Diaz Navas http://www.ugr.es/~jadiaz/                 +         + ++= π λ π f yx dIIIII 22 2121 2cos 2 cos2 I – intensity of the interference fringes I1, I2 – intensity of the intensities of the two beams λ – wavelength d – 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 and Philos. Mag. (5) 13 (1892), pp.236-242
31 SOLO Michelson Interferometer – Broad Source Interference 1882 Nobel Prize 1907 José Antonio Diaz Navas http://www.ugr.es/~jadiaz/ [ ]                 +         +                     +∆ −++= π λ π υπ f yx d Logc yx f d IIIII 22 22 2121 2cos 2 cos 2 1 cos exp2 I – intensity of the interference fringes I1, I2 – intensity of the intensities of the two beams λ, c – wavelength, speed of light d – path length difference between the two interferometers arms x,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 Abraham Michelson 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 light and Maxwell’s Equations but inconsistent with Galilean Relativity. SOLO 1887 Return to Table of Content
33 Interferometers SOLO Ernst Mach and Ludwig Zehnder separately described what has become the Mach-Zehnder Interferometer. 1891/92 Ludwig Louis Albert Zehnder 1854-1949 Ernst Waldfried Joseph Wenzel Mach 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 Return to Table of Content
34 InterferenceSOLO Sirks-Pringsheim Interferometer 1893/1898 J. A. Sirks, Hd. Ned. Nat. en Geneesk. Congr., Groningen, p. 92, 1893 E. 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
35 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 refractive index of a gas. His refractometer is based on Young’s double slit 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 equal thickness are rotated by the single knob D. One path length is shorted the other lengthened to compensate the difference between the paths. S 1S 2 S 2 T 1 T f 1C 2 C D D 1C 2C E Rayleigh's Interferometer Return to Table of Content
36 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 Return to Table of Content
3737 Optics History SOLO 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 Return to Table of Content
38 InterferenceSOLO Twyman-Green Interferometer 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 Return to Table of Content
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 Return to Table of Content
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. Return to Table of Content
41 InterferenceSOLO Hanbury-Brown and Twiss Interferometer 1956 Robert Hanbury-Brown and Richard Q. Twiss published “A test of a new type of stellar interferometer on Sirius”, Nature, vol. 178, pp.1046, 1956 Richard Q. Twiss 1920 - 2005 Robert Hanbury-Brown 1916 - 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. Return to Table of Content
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 back mirror has a high reflectivity. If no losses occur in the cavity, the power reflectivity is unity at all wavelength, but the phase of the reflected light is frequency-dependent due to the cavity effect, causing group delay dispersion (GDD). Return to Table of Content
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 ( ) ( ) ( )[ ]{ } ( ){ }tUtiAtAtu 222222 ReexpRecos =+=+= φωφω ( ) ( )[ ]222 exp: φω += tiAtU 1 U 2 U 21 UUU += 1φ2φ φ ( )1221 2 2 2 1 212211 cos2 2 φφ −⋅⋅++= ⋅⋅+⋅+⋅== ∗∗∗ AAAA UUUUUUUA 2 U ( )       + + = −++= − 2211 22111 2121 2 2 2 1 coscos sinsin tan cos2 φφ φφ φ φφ AA AA AAAAA The Phasor summation is identical to Vector 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  1 S 2 S P 1r  2 r  2211 1 2 :&1 2 : rkrk λ π λ π ==  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 1 r  2 r  The Irradiance at the point P is given by: ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )trEtrEtrEtrEtrEtrEtrEtrE trEtrEtrEtrEtrEtrEI ,,,,,,,, ,,,,,, 1122221122221111 22112211   ∗∗∗∗ ∗∗∗ ⋅+⋅+⋅+⋅= +⋅+=⋅= εεεε εε ( ) ( ) ( ) ( )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 1 S 2 S P 1 r  2 r  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 1 2 :&1 2 : rkrk λ π λ π ==  Since ,2,1,0& 2 2 21 122121max ±±== − +−←++= mmrrIIIII λλ π φφ ( ) ,2,1,0& 2 12 2 2 21 122121min ±±=+= − +−←−+= mmrrIIIII λ λ π φφ The Visibility of the fringes is defined as: 21 21 minmax minmax 2 : II II II II V + = + − = Return to Table of Content
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 optical devices 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 Return to Table of Content
49 Wavefront-Splitting InterferometerSOLO Young’s Experiment 1r 2 r s a y 2S 1S P OS 'O aΣ oΣ Young passed sun light through a pinhole, which become the primare source, obtained a spatially coherent beam through two identically illuminated apertures. The two apertures acted as two coherent sources producing a system of alternating bright and dark bands of interference fringes. Given a point P on the screen at distances r1 and r2 from apertures S1 and S2, respectively. We have The path difference is: 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== ( ) ( ) ( ) s a y s z s ay s z s ay s zy a szy a srr ays =               + − −−      + + +≈ +      −+−+      ++=− +>> 2 2 2 2 2 2 2 2 2/ 2 2 22 2 2 12 2/ 2 1 1 2/ 2 1 1 22 http://homepage.univie.ac.at/Franz.Embacher/KinderUni2005/waves.gif 2S 1S y z
50 SOLO Young’s Experiment (continue – 1) s a yrr sa<< ≈− 12 The bright fringes are obtained when: ,2,1,012 ±±==− mmrr λ ,2,1,0 ±±== mm s a y λ The distance between two consecutive bright fringes is: ( ) λλλ a s a s m a s myyy mm =−+≈−=∆ + 11 The dark fringes are obtained when: ,2,1,0 2 12 ±±=+=− mmrr λ λ ( ) ,2,1,0 2 12 ±±=+= mm s a y λ λ - wavelength The Intensity at point P is: ( ) ( )[ ]{ }       =−+=−+⋅−⋅++= = −=− == = s ya IrrkIrkrkIIIII k syarr III λ π φφ λπ φφ 2 0 /2 / 1202111222121 cos4cos12cos2 12 021 21  1r 2r s a y 2S 1S P OS 'O aΣ oΣ 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 of two planar mirrors inclined to each other at a very small angle δ. Wavefront-Splitting Interferometer Augustin Jean Fresnel 1788-1827 Fresnel’s Double Mirror (1819*)
52 SOLO Fresnel’s Double Mirror (continue – 1) Fresnel’s Double Mirror consists of two planar mirrors inclined to each other at a very small angle δ. The slit S image of the first mirror is S1 and of the second mirror is S2. The points S, S1 and S2 determine a plane normal to both planar mirrors that intersects them at a point C (on the intersection line of the two mirrors) We have: RCSCSSC === 21 δ=∠ 21 SSS Since is normal to the first mirror and is normal to second mirror, we have: 1SS 2 SS Also: δ⋅=∠⋅=∠ 22 2121 SSSSCS a 1 S 2S S R R R Screen Schield Mirror 2 C δ δ δ δ s Mirror 1 S C I C We will arrange a planar screen perpendicular to the normal from point C to line, , that also bisects the angle .aSS =21 δ221 =∠ SCSCCI – is the distance between line and the screen.21 SSSI CCs = A shield is introduced to prevent the waveform to travel straight from slit S to Wavefront-Splitting Interferometer
53 SOLO Fresnel’s Double Mirror (continue – 2) From the slit S a cylindrical waveform is reflected by one side of the mirror at point A and reaches the screen at point P, while an other cylindrical waveform is reflected by the other side of the mirror at point B and interferes with the first at the point P on the screen. Because of the reflection: BSSBASSA 21 & == Therefore we have: 111 rPSAPASAPSA ==+=+ 222 rPSBPBSBPSB ==+=+ a 1S 2S S A B P R R R Screen Schield Mirror 2 C δ δ δ δ s 2 r 1r Mirror 1 y S C I C where: – is the distance between line and the screen.21 SSSI CCs = PCySSa S == &21 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== Wavefront-Splitting Interferometer
54 S P R Screen Schield Mirror2 C δ s Mirror1 Slit S C y z SOLO Fresnel’s Double Mirror (continue – 3) We have: The path difference is: The bright fringes are obtained when: ( ) λλλ a s a s m a s myyy mm =−+≈−=∆ + 11 a 1S 2S S A B P R R R Screen Schield Mirror 2 C δ δ δ δ s 2 r 1 r Mirror 1 y S C I C S A B P R Screen Schield Mirror2 C δ s Mirror1 Slit SC y z S A B P R Screen Schield Mirror2 C δ s Mirror1 Slit y SC z P 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== ( ) ( ) ( ) s a y s z s ay s z s ay s zy a szy a srr ays =               + − −−      + + +≈ +      −+−+      ++=− +>> 2 2 2 2 2 2 2 2 2/ 2 2 22 2 2 12 2/ 2 1 1 2/ 2 1 1 22 Wavefront-Splitting Interferometer ,2,1,0& 2 21 12 ±±== − +− mmrr λλ π φφ Since the distance between two consecutive bright fringes is: 21 φφ = Classes of Interferometers Return to Table of Content
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 from a 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 1 S O S 'O aΣ o Σ 1<<α iθ d iθ - incident angle δ - dispersion angle α - prism angle From the Figure we can see that two virtual sources S1 and S2 exists. Let a be the distance between them. From the Figure ( ) δδθθ θ δ ddd a i ii 1 1 sintan 2 << << ≈−−= where θi – ray incident angle δ – ray dispersion (deviation) angle d – distance slit to bi-prism vertex α – prism angle ( )[ ] [ ] ( )ααθαθ αθαθαθδ α θ α θ 1sin sincossinsinsin 1 1 1 1 1 2/1221 −≈−−+≈ −−−+= << << − << << − nn n n ii iii ii See δ development Wavefront-Splitting Interferometer
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 1 sin it n θθ = 10 ≈n ( )[ ]1 2/1 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 a between virtual sources S1 and S2 is: ( ) δδθθ θ δ ddd a i ii 1 1 sintan 2 << << ≈−−= ( )[ ] [ ] ( )ααθαθ αθαθαθδ α θ α θ 1sin sincossinsinsin 1 1 1 1 1 2/1221 −≈−−+≈ −−−+= << << − << << − nn n n ii iii ii See δ development ( )α12 −≈ nda s a y 2S 1 S P OS 'O aΣ o Σ 1<<α α - prism angle 1r 2r Screen Bi-prism Slit y z Consider two rays starting from the slit S that pass 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, and having optical paths r1 and r2, respectively. 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== Wavefront-Splitting Interferometer δ s a 2S 1 S O S 'O a Σ o Σ 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 1 S P OS 'O a Σ o Σ 1<<α α - prism angle 1r 2 r d Screen Bi-prism Slit y z 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== The path difference is: The bright fringes are obtained when: ( ) λλλ a s a s m a s myyy mm =−+≈−=∆ + 11 ( ) ( ) ( ) s a y s z s ay s z s ay s zy a szy a srr ays =               + − −−      + + +≈ +      −+−+      ++=− +>> 2 2 2 2 2 2 2 2 2/ 2 2 22 2 2 12 2/ 2 1 1 2/ 2 1 1 22 We have: Wavefront-Splitting Interferometer ,2,1,0& 2 21 12 ±±== − +− mmrr λλ π φφ Since the distance between two consecutive bright fringes is: 21 φφ = Classes of Interferometers Return to Table of Content
59 SOLO Lloyd’s Mirror Interferometer The Lloyd’s planar mirror is perpendicular to the planar screen. A cylindrical waveform from the slit S is reflected by the mirror and interferes at the screen with the portion of the wave that proceeds directly to the screen. Screen Plane Mirror Slit y z From the Figure we can see that a virtual source S1, that is symmetric relative to mirror plane exists. The slit, parallel to mirror plane, is at the same distance, a/2, from the mirror plane as it’s virtual image. Wavefront-Splitting Interferometer sa y 1S P O S o Σ 1r 2 r2/a 2/a Planar Mirror Screen Screen Plane Mirror Slit y z Consider two rays starting from the slit S, one proceeding directly to the screen and the other reflected 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, and having optical paths r1 and r2, respectively. 2 2 2 12 2 2 2 1 22 zy a sPSrzy a sPSr +      ++==+      −+== Humphrey Lloyd 1800-1881
60 SOLO Lloyd’s Mirror Interferometer (continue – 1) Wavefront-Splitting Interferometer sa y 1S P O S oΣ 1r 2r2/a 2/a Planar Mirror Screen Screen Plane Mirror Slit y z The path difference is: The bright fringes are obtained when: The distance between two consecutive bright fringes is: ( ) λλλ a s a s m a s myyy mm =−+≈−=∆ + 11 ( ) ( ) ( ) s a y s z s ay s z s ay s zy a szy a srr ays =               + − −−      + + +≈ +      −+−+      ++=− +>> 2 2 2 2 2 2 2 2 2/ 2 2 22 2 2 12 2/ 2 1 1 2/ 2 1 1 22 We have: 2 2 2 12 2 2 2 1 22 zy a sPSrzy a sPSr +      ++==+      −+== ,2,1,0& 2 21 12 ±±== − +− mmrr λλ π φφ Classes of Interferometers Return to Table of Content
61 Stokes Treatment of Reflection and RefractionSOLO 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 RefractionSOLO 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' θθ θθθθ 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 11 2 coscos cos221 θθ θ θ µµ nn n t + = = ⊥ 2112 11 || coscos cos221 θθ θµµ nn n t + = = ( ) 2211 2211 1 coscos coscos21 θθ θθ θ µµ nn nn r + − = = ⊥ ( ) 2112 2112 1|| coscos coscos21 θθ θθ θ µµ nn nn r + − = = Parallel Interfaces Return 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 slab that performs a double reflection giving two coherent rays (1) and (2). Using a lens the two rays interfere at lens focus. 'D 1 θ 1 θ 1 θ 2θ 2 θ d C B D 1n 2 n 1 n Point source Image 1 2 Dielectric slab 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 1 22 '2 exp' λ π ω BDn tiADE For the reflected ray (2) we have at point D’. DD’ is normal two ray (2) so that both rays travel the same optical paths until interference. ( ) ( )         + −= 0 2 11 2 exp λ π ω 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 : 'D 1 θ 1 θ 1θ 2θ 2 θ d C B D 1 n 2n 1 n Point source Image 1 2 Dielectric slab 2θ 2 θ 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 1 2 21 1 2 << ≈−= θ θθτθτ r 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’. Amplitude Split Interferometers
65 SOLO Optical Reflected Path Length Difference: Parallel Interfaces (continue – 2) Two-Beam Interference: Parallel Interfaces 'D 1 θ 1 θ 1 θ 2θ 2 θ d C B D 1n 2 n 1 n Point source Image 1 2 Dielectric slab ( ) ( )         −= 0 1 12 '2 exp' λ π ωθ BDn tirADE ( ) ( ) ( )         + + −= π λ π ωθ 0 2 11 2 exp CDBCn tirADE 2cos/ θdCDBC == From the Figure we obtain: 12 sintan2' θθdBD = The phase difference at interference is: ( )[ ] π λ π φφ +−+−=− BDnCDBCn 12 0 21 2 2 2 2 2 sinsin 1 2 2 1121 cos sin sin cos sin sintan 2211 θ θ θ θ θ θθ θθ nnn nn = == ( ) πθ λ π πθ θλ π πθθ θλ π φφ +−=+      −−=+               −−=− 2 0 2 2 2 2 2 0 121 2 2 0 21 cos 4 sin1 cos 22 sintan cos 2 2 ndnd n n d Amplitude Split Interferometers
66 SOLO Optical Reflected Path Length Difference: Parallel Interfaces (continue – 3) Two-Beam Interference: Parallel Interfaces 'D 1 θ 1 θ 1 θ 2 θ 2 θ d C B D 1 n 2 n 1n Point source Image 1 2 Dielectric slab ( ) ( ) ( )212 exp' φωθ += tirADE ( ) ( ) ( )111 exp φωθ += tirADE πθ λ π φφ +−=− 2 0 2 21 cos 4 nd The Intensity at the interference is: ( ) ( ){ } ( ){ } ( )2/sin4 cos12cos12 cos2 2 0 0210 2111222121 021 122 δ πδφφ φφ I II rkrkIIIII III rkrk = +−+=−+= −+⋅−⋅++= == ⋅=⋅   where 2 0 2 cos 4 : θ λ π δ nd = ( )1 22 021 ~ θrAIII == Amplitude Split Interferometers Return to Table of Content
67 SOLO Optical Transmitted Path Length Difference: Parallel Interfaces Two-Beam Interference: Parallel Interfaces Amplitude Split Interferometers ( ) ( ) ( )         + −= 0 12 12 '2 exp' λ π ωθ BDnABn tirADE ( ) ( ) ( )         + ++ −= π λ π ωθ 0 2 11 2 exp CDBCABn tirADE 2cos/ θdCDBCAB === From the Figure we obtain: 12 sintan2' θθdBD = The phase difference at interference is: ( )[ ] π λ π φφ +−+−=− ' 2 12 0 21 BDnCDBCn 2 2 2 2 sinsin 1 2 2 1121 cos sin sin cos sin sintan 2211 θ θ θ θ θ θθ θθ nnn nn = == ( ) πθ λ π πθ θλ π πθθ θλ π φφ +−=+      −−=+               −−=− 2 0 2 2 2 2 2 0 121 2 2 0 21 cos 4 sin1 cos 22 sintan cos 2 2 ndnd n n d ( ) ( ) ( ) ( )01 0 2 10 exp 2 exp δωθ λ π ωθ −=        −= tirA ABn tirABE 2 0 2 20 2 0 cos 4 : cos 2 : θ λ π δ θλ π δ nd nd = = Return to Table of Content
68 InterferenceSOLO Haidinger Fringes 1846 Wilhelm Karl, Ritter von Haidinger 1795 - 1871 Lens Beam- splitter Extended Sources Viewing Screen Dielectric film Black background Circular fringes Haidinger Fringes are the type of interference pattern that results with an extended source where partial reflections occur from a plane-parallel dielectric slab.
69 SOLO Haidinger Interference Fringes Two-Beam Interference: Parallel Interfaces We have: 1θ 1θ 2 θ 2 θ d 1n 2 n 1n Extended source Focal plane 1P 2 P 1θ Dielectric slab Beam splitter Lens θ f x Haidinger Fringes are the type of interference pattern that results with an extended source where partial reflections occur from a plane-parallel dielectric slab. Wilhelm Karl, Ritter von Haidinger 1795 - 1871 Amplitude Split Interferometers Return to Table of Content
70 InterferenceSOLO Interference of Many Monochromatic Waves Given two waves ( ω = constant ): ( ) ( ) ( )[ ]{ } ( ){ }tUtiAtAtu 111111 ReexpRecos =+=+= φωφω The N waves interfere to give: ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } ( )φω +=+++= +++= tAtUtUtU tutututu N N cosRe 21 21   ( ) ( ) ( )[ ]{ } ( ){ }tUtiAtAtu 222222 ReexpRecos =+=+= φωφω 1U N UUUU +++= 21 1φ 2φ φ 2 U N U 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 1 0 32 2 0 3 3 02 01 '' '' '' −−− − − = = = = NtiN rN ti r ti r ti r 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 1 0 12 2 0 4 3 0 2 2 01 0 0 0 0 '' '' '' ' −−−− −− −− − = = = = NtiiN rN tii t tii t tii t eeEtrtE eeEtrtE eeEtrtE eeEttE 2 0 2 20 2 0 cos 4 : cos 2 : θ λ π δ θλ π δ nd nd = =
73 InterferenceSOLO Multiple Beam Interference from a Parallel Film Using lens the multi-rays interfere at lens focus. ( ) [ ] tiNiNii rNrrr eEetrtetrtetrtr EEEE ωδδδ 0 13223 21 ''''''   +++++= ++++= −−−−− ( ) ti i NiN i eE er er etrtr ω δ δ δ 02 132 '1 '1 ''       − − += − −−− − ∞→ < Nand rIf 1' ti i i r eE er etrt rE ω δ δ 02 '1 ''       − += − − In the case of zero absorption, no energy being taken out of the waves, using Stokes relations 2 1'&' rttrr −=−= ( ) ti i i r eE er er E ω δ δ 02 1 1       − − = − − ∞→ < Nand rIf 1
74 InterferenceSOLO Multiple Beam Interference from a Parallel Film Using lens the multi-rays interfere at lens focus. ( ) ( ) [ ] ( )0 0 112242 21 ''''1 δωδδδ −−−−−− +++++= ++++= tiNiNii tNttt eEttererer EEEE   ( )0 02 2 ' '1 '1 δω δ δ − − − − − = ti i NiN eEtt er er ∞→ < Nand rIf 1' ( )0 02 '1 ' δω δ − − − = ti it eE er tt E In the case of zero absorption, no energy being taken out of the waves, using Stokes relations 2 1'&' rttrr −=−= ( )0 02 2 1 1 δω δ − − − − = ti it eE er r E 2 0 2 20 2 0 cos 4 : cos 2 : θ λ π δ θλ π δ nd nd = =
75 InterferenceSOLO Multiple Beam Interference from a Parallel Film ∞→ < Nand rIf 1 ( ) ti i i r eE er er E ω δ δ 02 1 1       − − = − − ( )0 02 2 1 1 δω δ − − − − = ti it eE er r E Let compute the Reflected and Transmitted Irradiances: ( ) ( ) ( ) ( ) 024 2 * 0022 * cos21 cos12 1 1 1 1 I rr r EE er er er er EEI i i i i rrr δ δ δ δ δ δ −+ − =      − − − − =∝ − − ( ) ( ) 024 22 * 002 2 2 2 * cos21 1 1 1 1 1 I rr r EE er r er r EEI iittt δδδ −+ − = − − − − =∝ − Using lens the multi-rays interfere at lens focus we found that in the case of zero absorption, no energy being taken out of the waves, using Stokes relations 2 1'&' rttrr −=−= 0III tr =+ ( ) ( )[ ] ( ) ( )[ ] ( ) 0222 2222/sin21cos 2/sin1/21 2/sin1/2 2 I rr rr Ir δ δδδ −+ − = −= ( ) ( )[ ] ( ) 0222 2/sin21cos 2/sin1/21 1 2 I rr It δ 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 Multiple Beam Interference from a Parallel Film Return to Table of Content
77 Gas RefrectometerSOLO S 1S 2 S 2T 1T f 1C 2 C D 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 1 T 1 C 2 C D D 1C 2C E 1 G 2 G 1 2 Jamin's Interferometer t S 1 T E 1 M 2 G 1 2 Mach-Zehnder Interferometer 2M 3 M 4M 2T 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 3 1, 2 2 n nVN Tpng Reyleigh’s Interferometer Jamin’s Interferometer Return to Table of Content
78 Interferometers History SOLO References 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/ Return to Table of Content
January 5, 2015 79 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
80 J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
81J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
82J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
83J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
84
85
86
87
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 Interferometer Mach-Zehnder Interferometer Fabry-Perot Interferometer Sagnac Interferometer Gires-Tournois Etalon
89 Field and linear interferometers InterferenceSOLO Moire Interferometer (see Moire pattern) Holographic Interferometer Near-field Interferometer Fringes 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 http://en.wikipedia.org/wiki/List_of_types_of_interferometers Fizeau Interferometer Rayleigh Interferometer Twyman-Green Interferometer
90 Intensity and nonlinear interferometers InterferenceSOLO http://en.wikipedia.org/wiki/List_of_types_of_interferometers Intensity Interferometer Intensity Optical Correlator Frequency-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 Interferometer Ramsey Interferometer Mini 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.
93

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Interferometers history

  • 1.
  • 2.
    2 InterferenceSOLO Table of Content Introduction HaidingerFringes 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 Interferometer Gires-Tournois Etalon
  • 3.
    3 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 Experiment Division of Wavefront Optical Reflected Path Length Difference: Parallel Interfaces Amplitude Split Interferometers Stokes Treatment of Reflection and Refraction Optical Transmitted Path Length Difference: Parallel Interfaces Haidinger Interference Fringes Interference of Many Monochromatic Waves Gas Refrectometer References
  • 4.
    4 InterferenceSOLO Introduction Interference is thesuperposition 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.
    5 SOLO Diffraction The Grimaldi’sdescription of diffraction was published in 1665 , 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 the Jesuit college in Bolognia discovered the diffraction of light and gave it the name diffraction, 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 be which is formed by the direct rays. Furter it should not be omitted that the illuminated base IK appears in the middle suffused with pure light, and either extremity its light is colored.” Single Slit Diffraction Double Slit Diffraction http://en.wikipedia.org/wiki/Francesco_Maria_Grimaldi 1665
  • 6.
    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 compound microscope: on the left the illumination device (an oil lamp), on the 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-front derivation 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.
    7 OpticsSOLO Newton published “Opticks” 1704 Inthis 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 the corpuscular theory. This conviction was due to the fact 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.
  • 8.
  • 9.
    9 OpticsSOLO History (continue) In 1801Thomas Young uses constructive and destructive interference of waves to explain the Newton’s rings. Thomas Young 1773-1829 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.
    10 POLARIZATIONSOLO History In 1802 WilliamHyde Wollaston discovered that the sun spectrum is composed by a number of dark lines, but the interpretation of this phenomenon was done by Fraunhofer in 1814. Wollaston developed In 1802 the refractometer, an instrument used to measure the refractive index. 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, used in microscopy. 1802-1807
  • 11.
    11 SOLO History In 1813 JosephFraunhofer 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 light passing through this combination with a small telescope eypiece. By this technique he was able to investigate the spectrum bit by bit, color by color.
  • 12.
    12 SOLO Between 1805 and1815 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. Diffraction http://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.”
  • 13.
    13 SOLO In 1818 AugustFresnel 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. Diffraction http://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 Arago 1786-1853 Siméon Denis Poisson 1781-1840 Pierre-Simon Laplace (1749-1827) Joseph Louis Guy-Lussac 1778-1850 Judging Committee of French Academy 1818
  • 14.
    14 SOLO Diffraction http://microscopy.fsu.edu/optics/timeline/people/gregory.html http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html Dominique François JeanArago 1786-1853 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 Poisson 1781-1840 Poisson’s or Arago’s Spot 1818
  • 15.
    15 POLARIZATION Arago and Fresnelinvestigated the interference of polarized rays of light and found in 1816 that two rays polarized at right angles to each other never interface. SOLO History (continue) Dominique François Jean Arago 1786-1853 Augustin Jean Fresnel 1788-1827 Arago relayed to Thomas Young in London the results of the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillations in 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 longitudinal waves transversal waves
  • 16.
    16 DiffractionSOLO History of Diffraction AugustinJean Fresnel 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
  • 17.
    17 DiffractionSOLO Augustin Jean Fresnel 1788-1827 In 1818Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves. P 0P Q 1x 0x 1y 0y η ξ Fr  Sr  ρ  r  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 source Q e r A tU − = ' ' ω According to Huygens Principle second wavelets will start at the aperture and will add at the image point P. ( ) ( ) ( )( ) ( ) ( )( ) ∫∫ Σ ++− Σ +−− == dre rr A Kdre r U KtU rrktj S sourcerkttjQ P S 2/2/' ',', πωπω θθθθ where: ( )',θθK obliquity or inclination factor ( ) ( )SSS nrnr 11cos&11cos' 11 ⋅=⋅= −− θθ ( ) ( )   === === 0',0 max0',0 πθθ θθ 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:
  • 18.
    18 SOLO History In 1821 JosephFraunhofer build the first diffraction grating, made up of 260 close parallel wires. Latter he built a diffraction grating 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.
    19 SOLO ELECTROMAGNETICS Conical Refractionon 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 monochromatic unpolarized light is normal to one of the crystal faces, the energy will spread out in the plate in a hollow cone, the cone 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 Hamilton in 1832 and confirmed experimentally by Humphrey Lloyd, a year later (Born & Wolf). Because it is no easy to obtain an accurate parallel beam of monochromatic light on obtained two bright circles (Born & Wolf). 1832 William Rowan Hamilton (1805-1855) Humphrey Lloyd (1800-1881)
  • 20.
    20 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
  • 21.
    21 InterferenceSOLO Haidinger Fringes 1846 Wilhelm Karl, Rittervon Haidinger 1795 - 1871 Lens Beam- splitter Extended Sources Viewing Screen Dielectric film Black background Circular fringes Haidinger Fringes are the type of interference pattern that results with an extended source where partial reflections occur from a plane-parallel dielectric slab. Return to Table of Content
  • 22.
    22 SOLO Fizeau Experiment Armand HyppoliteLouise 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 of light passes at the opening between teeth passes through L2 and is returned by the spherical mirror back to the toothed wheel. The rotation speed of the wheel is adjusted such 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 =⋅= skm t d v /000,311 000,18/1 633,822 = ⋅ == 1849 Return to Table of Content
  • 23.
    23 InterferenceSOLO Jamin’s Interferometer 1856 J. Jamin,C.R. Acad. Sci. Paris, 42, p.482, 1856 Jules Célestin Jamin 1818 - 1886 S 2 T 1 T 1C 2C D D 1 C 2C E 1G 2 G 1 2 Jamin's Interferometer In the Jamin’s Interferometer a monochromatic light 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 as function of presure and temperature, two identical empty tubes T1 and T2 are placed in the two parallel 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 equal thickness are rotated by the single knob D. One path length is shorted the other lengthened to compensate the difference between the paths. Return to Table of Content
  • 24.
    24 SOLO Fizeau Experiment inMoving 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 the half 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 1 1 n uvv 1859
  • 25.
  • 26.
  • 27.
    27 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 Return to Table of Content
  • 28.
    28 SOLO Foucault Experiment Foucault workingwith Fizeau improved the apparatus by replacing the toothed wheel with a rotating mirror. In 1850 Foucault used the improved apparatus to measure the speed of light in air and in water. In 1862 he used an improved version to give an accurate measurement of 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 the ray 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.
    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. Return to Table of Content
  • 30.
    30 SOLO Michelson Interferometer –Interference Fringes Interference 1882 Nobel Prize 1907 José Antonio Diaz Navas http://www.ugr.es/~jadiaz/                 +         + ++= π λ π f yx dIIIII 22 2121 2cos 2 cos2 I – intensity of the interference fringes I1, I2 – intensity of the intensities of the two beams λ – wavelength d – 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 and Philos. Mag. (5) 13 (1892), pp.236-242
  • 31.
    31 SOLO Michelson Interferometer –Broad Source Interference 1882 Nobel Prize 1907 José Antonio Diaz Navas http://www.ugr.es/~jadiaz/ [ ]                 +         +                     +∆ −++= π λ π υπ f yx d Logc yx f d IIIII 22 22 2121 2cos 2 cos 2 1 cos exp2 I – intensity of the interference fringes I1, I2 – intensity of the intensities of the two beams λ, c – wavelength, speed of light d – path length difference between the two interferometers arms x,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.
    32 SPECIAL RELATIVITY Michelson andMorley Experiment Albert Abraham Michelson 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 light and Maxwell’s Equations but inconsistent with Galilean Relativity. SOLO 1887 Return to Table of Content
  • 33.
    33 Interferometers SOLO Ernst Mach andLudwig Zehnder separately described what has become the Mach-Zehnder Interferometer. 1891/92 Ludwig Louis Albert Zehnder 1854-1949 Ernst Waldfried Joseph Wenzel Mach 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 Return to Table of Content
  • 34.
    34 InterferenceSOLO Sirks-Pringsheim Interferometer 1893/1898 J. A.Sirks, Hd. Ned. Nat. en Geneesk. Congr., Groningen, p. 92, 1893 E. 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
  • 35.
    35 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 refractive index of a gas. His refractometer is based on Young’s double slit 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 equal thickness are rotated by the single knob D. One path length is shorted the other lengthened to compensate the difference between the paths. S 1S 2 S 2 T 1 T f 1C 2 C D D 1C 2C E Rayleigh's Interferometer Return to Table of Content
  • 36.
    36 InterferenceSOLO Fabry – PerotInterferometer 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 Return to Table of Content
  • 37.
    3737 Optics History SOLO 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 Return to Table of Content
  • 38.
    38 InterferenceSOLO Twyman-Green Interferometer 1916 T. Twymanand 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 Return to Table of Content
  • 39.
    39 SOLO Michelson’s Experiments Michelson performeda 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 Return to Table of Content
  • 40.
    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. Return to Table of Content
  • 41.
    41 InterferenceSOLO Hanbury-Brown and TwissInterferometer 1956 Robert Hanbury-Brown and Richard Q. Twiss published “A test of a new type of stellar interferometer on Sirius”, Nature, vol. 178, pp.1046, 1956 Richard Q. Twiss 1920 - 2005 Robert Hanbury-Brown 1916 - 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. Return to Table of Content
  • 42.
    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 back mirror has a high reflectivity. If no losses occur in the cavity, the power reflectivity is unity at all wavelength, but the phase of the reflected light is frequency-dependent due to the cavity effect, causing group delay dispersion (GDD). Return to Table of Content
  • 43.
    43 InterferenceSOLO Interference of TwoMonochromatic 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 ( ) ( ) ( )[ ]{ } ( ){ }tUtiAtAtu 222222 ReexpRecos =+=+= φωφω ( ) ( )[ ]222 exp: φω += tiAtU 1 U 2 U 21 UUU += 1φ2φ φ ( )1221 2 2 2 1 212211 cos2 2 φφ −⋅⋅++= ⋅⋅+⋅+⋅== ∗∗∗ AAAA UUUUUUUA 2 U ( )       + + = −++= − 2211 22111 2121 2 2 2 1 coscos sinsin tan cos2 φφ φφ φ φφ AA AA AAAAA The Phasor summation is identical to Vector summation
  • 44.
    44 InterferenceSOLO Interference of MonochromaticWaves 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  1 S 2 S P 1r  2 r  2211 1 2 :&1 2 : rkrk λ π λ π ==  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.
    45 InterferenceSOLO Interference of MonochromaticWaves 1S 2S P 1 r  2 r  The Irradiance at the point P is given by: ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )trEtrEtrEtrEtrEtrEtrEtrE trEtrEtrEtrEtrEtrEI ,,,,,,,, ,,,,,, 1122221122221111 22112211   ∗∗∗∗ ∗∗∗ ⋅+⋅+⋅+⋅= +⋅+=⋅= εεεε εε ( ) ( ) ( ) ( )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.
    46 InterferenceSOLO Interference of MonochromaticWaves 1 S 2 S P 1 r  2 r  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 1 2 :&1 2 : rkrk λ π λ π ==  Since ,2,1,0& 2 2 21 122121max ±±== − +−←++= mmrrIIIII λλ π φφ ( ) ,2,1,0& 2 12 2 2 21 122121min ±±=+= − +−←−+= mmrrIIIII λ λ π φφ The Visibility of the fringes is defined as: 21 21 minmax minmax 2 : II II II II V + = + − = Return to Table of Content
  • 47.
    47 InterferenceSOLO Billet’s Split Lens Meslin’sExperiment 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 optical devices 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.
    48 Two Basic Classesof Interferometers InterferenceSOLO Return to Table of Content
  • 49.
    49 Wavefront-Splitting InterferometerSOLO Young’s Experiment 1r 2 r s a y 2S 1S P OS'O aΣ oΣ Young passed sun light through a pinhole, which become the primare source, obtained a spatially coherent beam through two identically illuminated apertures. The two apertures acted as two coherent sources producing a system of alternating bright and dark bands of interference fringes. Given a point P on the screen at distances r1 and r2 from apertures S1 and S2, respectively. We have The path difference is: 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== ( ) ( ) ( ) s a y s z s ay s z s ay s zy a szy a srr ays =               + − −−      + + +≈ +      −+−+      ++=− +>> 2 2 2 2 2 2 2 2 2/ 2 2 22 2 2 12 2/ 2 1 1 2/ 2 1 1 22 http://homepage.univie.ac.at/Franz.Embacher/KinderUni2005/waves.gif 2S 1S y z
  • 50.
    50 SOLO Young’s Experiment (continue– 1) s a yrr sa<< ≈− 12 The bright fringes are obtained when: ,2,1,012 ±±==− mmrr λ ,2,1,0 ±±== mm s a y λ The distance between two consecutive bright fringes is: ( ) λλλ a s a s m a s myyy mm =−+≈−=∆ + 11 The dark fringes are obtained when: ,2,1,0 2 12 ±±=+=− mmrr λ λ ( ) ,2,1,0 2 12 ±±=+= mm s a y λ λ - wavelength The Intensity at point P is: ( ) ( )[ ]{ }       =−+=−+⋅−⋅++= = −=− == = s ya IrrkIrkrkIIIII k syarr III λ π φφ λπ φφ 2 0 /2 / 1202111222121 cos4cos12cos2 12 021 21  1r 2r s a y 2S 1S P OS 'O aΣ oΣ http://homepage.univie.ac.at/Franz.Embacher/KinderUni2005/waves.gif Wavefront-Splitting Interferometer Classes of InterferometersReturn to Table of Content
  • 51.
    51 SOLO http://info.uibk.ac.at//c/c7/c704/museum/en/details/optics/fresnel.html University of Innsbruck Fresnel’sDouble Mirror consists of two planar mirrors inclined to each other at a very small angle δ. Wavefront-Splitting Interferometer Augustin Jean Fresnel 1788-1827 Fresnel’s Double Mirror (1819*)
  • 52.
    52 SOLO Fresnel’s Double Mirror(continue – 1) Fresnel’s Double Mirror consists of two planar mirrors inclined to each other at a very small angle δ. The slit S image of the first mirror is S1 and of the second mirror is S2. The points S, S1 and S2 determine a plane normal to both planar mirrors that intersects them at a point C (on the intersection line of the two mirrors) We have: RCSCSSC === 21 δ=∠ 21 SSS Since is normal to the first mirror and is normal to second mirror, we have: 1SS 2 SS Also: δ⋅=∠⋅=∠ 22 2121 SSSSCS a 1 S 2S S R R R Screen Schield Mirror 2 C δ δ δ δ s Mirror 1 S C I C We will arrange a planar screen perpendicular to the normal from point C to line, , that also bisects the angle .aSS =21 δ221 =∠ SCSCCI – is the distance between line and the screen.21 SSSI CCs = A shield is introduced to prevent the waveform to travel straight from slit S to Wavefront-Splitting Interferometer
  • 53.
    53 SOLO Fresnel’s Double Mirror(continue – 2) From the slit S a cylindrical waveform is reflected by one side of the mirror at point A and reaches the screen at point P, while an other cylindrical waveform is reflected by the other side of the mirror at point B and interferes with the first at the point P on the screen. Because of the reflection: BSSBASSA 21 & == Therefore we have: 111 rPSAPASAPSA ==+=+ 222 rPSBPBSBPSB ==+=+ a 1S 2S S A B P R R R Screen Schield Mirror 2 C δ δ δ δ s 2 r 1r Mirror 1 y S C I C where: – is the distance between line and the screen.21 SSSI CCs = PCySSa S == &21 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== Wavefront-Splitting Interferometer
  • 54.
    54 S P R Screen Schield Mirror2 C δ s Mirror1 Slit S C y z SOLO Fresnel’s Double Mirror(continue – 3) We have: The path difference is: The bright fringes are obtained when: ( ) λλλ a s a s m a s myyy mm =−+≈−=∆ + 11 a 1S 2S S A B P R R R Screen Schield Mirror 2 C δ δ δ δ s 2 r 1 r Mirror 1 y S C I C S A B P R Screen Schield Mirror2 C δ s Mirror1 Slit SC y z S A B P R Screen Schield Mirror2 C δ s Mirror1 Slit y SC z P 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== ( ) ( ) ( ) s a y s z s ay s z s ay s zy a szy a srr ays =               + − −−      + + +≈ +      −+−+      ++=− +>> 2 2 2 2 2 2 2 2 2/ 2 2 22 2 2 12 2/ 2 1 1 2/ 2 1 1 22 Wavefront-Splitting Interferometer ,2,1,0& 2 21 12 ±±== − +− mmrr λλ π φφ Since the distance between two consecutive bright fringes is: 21 φφ = Classes of Interferometers Return to Table of Content
  • 55.
    55 SOLO Fresnel’s Double Prism TheFresnel’s Double Prism or Bi-prism consists of two thin prisms joined at their bases. A singlr cylindrical wave emerge from a 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 1 S O S 'O aΣ o Σ 1<<α iθ d iθ - incident angle δ - dispersion angle α - prism angle From the Figure we can see that two virtual sources S1 and S2 exists. Let a be the distance between them. From the Figure ( ) δδθθ θ δ ddd a i ii 1 1 sintan 2 << << ≈−−= where θi – ray incident angle δ – ray dispersion (deviation) angle d – distance slit to bi-prism vertex α – prism angle ( )[ ] [ ] ( )ααθαθ αθαθαθδ α θ α θ 1sin sincossinsinsin 1 1 1 1 1 2/1221 −≈−−+≈ −−−+= << << − << << − nn n n ii iii ii See δ development Wavefront-Splitting Interferometer
  • 56.
    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 1 sin it n θθ = 10 ≈n ( )[ ]1 2/1 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.
    57 SOLO Fresnel’s Double Prism(continue – 1) From the Figure we found that the distance a between virtual sources S1 and S2 is: ( ) δδθθ θ δ ddd a i ii 1 1 sintan 2 << << ≈−−= ( )[ ] [ ] ( )ααθαθ αθαθαθδ α θ α θ 1sin sincossinsinsin 1 1 1 1 1 2/1221 −≈−−+≈ −−−+= << << − << << − nn n n ii iii ii See δ development ( )α12 −≈ nda s a y 2S 1 S P OS 'O aΣ o Σ 1<<α α - prism angle 1r 2r Screen Bi-prism Slit y z Consider two rays starting from the slit S that pass 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, and having optical paths r1 and r2, respectively. 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== Wavefront-Splitting Interferometer δ s a 2S 1 S O S 'O a Σ o Σ 1<<α iθ d iθ - incident angle δ - dispersion angle α - prism angle
  • 58.
    58 SOLO Fresnel’s Double Prism(continue – 2) ( )α12 −≈ nda s a y 2S 1 S P OS 'O a Σ o Σ 1<<α α - prism angle 1r 2 r d Screen Bi-prism Slit y z 2 2 2 22 2 2 2 11 22 zy a sPSrzy a sPSr +      ++==+      −+== The path difference is: The bright fringes are obtained when: ( ) λλλ a s a s m a s myyy mm =−+≈−=∆ + 11 ( ) ( ) ( ) s a y s z s ay s z s ay s zy a szy a srr ays =               + − −−      + + +≈ +      −+−+      ++=− +>> 2 2 2 2 2 2 2 2 2/ 2 2 22 2 2 12 2/ 2 1 1 2/ 2 1 1 22 We have: Wavefront-Splitting Interferometer ,2,1,0& 2 21 12 ±±== − +− mmrr λλ π φφ Since the distance between two consecutive bright fringes is: 21 φφ = Classes of Interferometers Return to Table of Content
  • 59.
    59 SOLO Lloyd’s Mirror Interferometer TheLloyd’s planar mirror is perpendicular to the planar screen. A cylindrical waveform from the slit S is reflected by the mirror and interferes at the screen with the portion of the wave that proceeds directly to the screen. Screen Plane Mirror Slit y z From the Figure we can see that a virtual source S1, that is symmetric relative to mirror plane exists. The slit, parallel to mirror plane, is at the same distance, a/2, from the mirror plane as it’s virtual image. Wavefront-Splitting Interferometer sa y 1S P O S o Σ 1r 2 r2/a 2/a Planar Mirror Screen Screen Plane Mirror Slit y z Consider two rays starting from the slit S, one proceeding directly to the screen and the other reflected 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, and having optical paths r1 and r2, respectively. 2 2 2 12 2 2 2 1 22 zy a sPSrzy a sPSr +      ++==+      −+== Humphrey Lloyd 1800-1881
  • 60.
    60 SOLO Lloyd’s Mirror Interferometer(continue – 1) Wavefront-Splitting Interferometer sa y 1S P O S oΣ 1r 2r2/a 2/a Planar Mirror Screen Screen Plane Mirror Slit y z The path difference is: The bright fringes are obtained when: The distance between two consecutive bright fringes is: ( ) λλλ a s a s m a s myyy mm =−+≈−=∆ + 11 ( ) ( ) ( ) s a y s z s ay s z s ay s zy a szy a srr ays =               + − −−      + + +≈ +      −+−+      ++=− +>> 2 2 2 2 2 2 2 2 2/ 2 2 22 2 2 12 2/ 2 1 1 2/ 2 1 1 22 We have: 2 2 2 12 2 2 2 1 22 zy a sPSrzy a sPSr +      ++==+      −+== ,2,1,0& 2 21 12 ±±== − +− mmrr λλ π φφ Classes of Interferometers Return to Table of Content
  • 61.
    61 Stokes Treatment ofReflection and RefractionSOLO 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.
    62 Stokes Treatment ofReflection and RefractionSOLO 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' θθ θθθθ 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 11 2 coscos cos221 θθ θ θ µµ nn n t + = = ⊥ 2112 11 || coscos cos221 θθ θµµ nn n t + = = ( ) 2211 2211 1 coscos coscos21 θθ θθ θ µµ nn nn r + − = = ⊥ ( ) 2112 2112 1|| coscos coscos21 θθ θθ θ µµ nn nn r + − = = Parallel Interfaces Return to Table of Content
  • 63.
    63 SOLO Optical Reflected PathLength Difference: Parallel Interfaces Two-Beam Interference: Parallel Interfaces We have a point source and a dielectric slab that performs a double reflection giving two coherent rays (1) and (2). Using a lens the two rays interfere at lens focus. 'D 1 θ 1 θ 1 θ 2θ 2 θ d C B D 1n 2 n 1 n Point source Image 1 2 Dielectric slab 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 1 22 '2 exp' λ π ω BDn tiADE For the reflected ray (2) we have at point D’. DD’ is normal two ray (2) so that both rays travel the same optical paths until interference. ( ) ( )         + −= 0 2 11 2 exp λ π ω CDBCn tiADE Amplitude Split Interferometers
  • 64.
    64 SOLO Optical Reflected PathLength Difference: Parallel Interfaces (continue – 1) Two-Beam Interference: Parallel Interfaces To compute the amplitudes A1 and A2 we will use : 'D 1 θ 1 θ 1θ 2θ 2 θ d C B D 1 n 2n 1 n Point source Image 1 2 Dielectric slab 2θ 2 θ 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 1 2 21 1 2 << ≈−= θ θθτθτ r 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’. Amplitude Split Interferometers
  • 65.
    65 SOLO Optical Reflected PathLength Difference: Parallel Interfaces (continue – 2) Two-Beam Interference: Parallel Interfaces 'D 1 θ 1 θ 1 θ 2θ 2 θ d C B D 1n 2 n 1 n Point source Image 1 2 Dielectric slab ( ) ( )         −= 0 1 12 '2 exp' λ π ωθ BDn tirADE ( ) ( ) ( )         + + −= π λ π ωθ 0 2 11 2 exp CDBCn tirADE 2cos/ θdCDBC == From the Figure we obtain: 12 sintan2' θθdBD = The phase difference at interference is: ( )[ ] π λ π φφ +−+−=− BDnCDBCn 12 0 21 2 2 2 2 2 sinsin 1 2 2 1121 cos sin sin cos sin sintan 2211 θ θ θ θ θ θθ θθ nnn nn = == ( ) πθ λ π πθ θλ π πθθ θλ π φφ +−=+      −−=+               −−=− 2 0 2 2 2 2 2 0 121 2 2 0 21 cos 4 sin1 cos 22 sintan cos 2 2 ndnd n n d Amplitude Split Interferometers
  • 66.
    66 SOLO Optical Reflected PathLength Difference: Parallel Interfaces (continue – 3) Two-Beam Interference: Parallel Interfaces 'D 1 θ 1 θ 1 θ 2 θ 2 θ d C B D 1 n 2 n 1n Point source Image 1 2 Dielectric slab ( ) ( ) ( )212 exp' φωθ += tirADE ( ) ( ) ( )111 exp φωθ += tirADE πθ λ π φφ +−=− 2 0 2 21 cos 4 nd The Intensity at the interference is: ( ) ( ){ } ( ){ } ( )2/sin4 cos12cos12 cos2 2 0 0210 2111222121 021 122 δ πδφφ φφ I II rkrkIIIII III rkrk = +−+=−+= −+⋅−⋅++= == ⋅=⋅   where 2 0 2 cos 4 : θ λ π δ nd = ( )1 22 021 ~ θrAIII == Amplitude Split Interferometers Return to Table of Content
  • 67.
    67 SOLO Optical Transmitted PathLength Difference: Parallel Interfaces Two-Beam Interference: Parallel Interfaces Amplitude Split Interferometers ( ) ( ) ( )         + −= 0 12 12 '2 exp' λ π ωθ BDnABn tirADE ( ) ( ) ( )         + ++ −= π λ π ωθ 0 2 11 2 exp CDBCABn tirADE 2cos/ θdCDBCAB === From the Figure we obtain: 12 sintan2' θθdBD = The phase difference at interference is: ( )[ ] π λ π φφ +−+−=− ' 2 12 0 21 BDnCDBCn 2 2 2 2 sinsin 1 2 2 1121 cos sin sin cos sin sintan 2211 θ θ θ θ θ θθ θθ nnn nn = == ( ) πθ λ π πθ θλ π πθθ θλ π φφ +−=+      −−=+               −−=− 2 0 2 2 2 2 2 0 121 2 2 0 21 cos 4 sin1 cos 22 sintan cos 2 2 ndnd n n d ( ) ( ) ( ) ( )01 0 2 10 exp 2 exp δωθ λ π ωθ −=        −= tirA ABn tirABE 2 0 2 20 2 0 cos 4 : cos 2 : θ λ π δ θλ π δ nd nd = = Return to Table of Content
  • 68.
    68 InterferenceSOLO Haidinger Fringes 1846 Wilhelm Karl, Rittervon Haidinger 1795 - 1871 Lens Beam- splitter Extended Sources Viewing Screen Dielectric film Black background Circular fringes Haidinger Fringes are the type of interference pattern that results with an extended source where partial reflections occur from a plane-parallel dielectric slab.
  • 69.
    69 SOLO Haidinger Interference Fringes Two-BeamInterference: Parallel Interfaces We have: 1θ 1θ 2 θ 2 θ d 1n 2 n 1n Extended source Focal plane 1P 2 P 1θ Dielectric slab Beam splitter Lens θ f x Haidinger Fringes are the type of interference pattern that results with an extended source where partial reflections occur from a plane-parallel dielectric slab. Wilhelm Karl, Ritter von Haidinger 1795 - 1871 Amplitude Split Interferometers Return to Table of Content
  • 70.
    70 InterferenceSOLO Interference of ManyMonochromatic Waves Given two waves ( ω = constant ): ( ) ( ) ( )[ ]{ } ( ){ }tUtiAtAtu 111111 ReexpRecos =+=+= φωφω The N waves interfere to give: ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } ( )φω +=+++= +++= tAtUtUtU tutututu N N cosRe 21 21   ( ) ( ) ( )[ ]{ } ( ){ }tUtiAtAtu 222222 ReexpRecos =+=+= φωφω 1U N UUUU +++= 21 1φ 2φ φ 2 U N U Nφ The Phasor summation is identical to Vector summation ( ) ( ) ( )[ ]{ } ( ){ }tUtiAtAtu NNNNNN ReexpRecos =+=+= φωφω
  • 71.
    71 InterferenceSOLO Multiple Beam Interferencefrom a Parallel Film We have a point source and a dielectric slab that performs a multiple reflection and transmission.
  • 72.
    72 InterferenceSOLO Multiple Beam Interferencefrom a Parallel Film ( ) ( ) ( )[ ]   δω δω δω ω 1 0 32 2 0 3 3 02 01 '' '' '' −−− − − = = = = NtiN rN ti r ti r ti r 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 1 0 12 2 0 4 3 0 2 2 01 0 0 0 0 '' '' '' ' −−−− −− −− − = = = = NtiiN rN tii t tii t tii t eeEtrtE eeEtrtE eeEtrtE eeEttE 2 0 2 20 2 0 cos 4 : cos 2 : θ λ π δ θλ π δ nd nd = =
  • 73.
    73 InterferenceSOLO Multiple Beam Interferencefrom a Parallel Film Using lens the multi-rays interfere at lens focus. ( ) [ ] tiNiNii rNrrr eEetrtetrtetrtr EEEE ωδδδ 0 13223 21 ''''''   +++++= ++++= −−−−− ( ) ti i NiN i eE er er etrtr ω δ δ δ 02 132 '1 '1 ''       − − += − −−− − ∞→ < Nand rIf 1' ti i i r eE er etrt rE ω δ δ 02 '1 ''       − += − − In the case of zero absorption, no energy being taken out of the waves, using Stokes relations 2 1'&' rttrr −=−= ( ) ti i i r eE er er E ω δ δ 02 1 1       − − = − − ∞→ < Nand rIf 1
  • 74.
    74 InterferenceSOLO Multiple Beam Interferencefrom a Parallel Film Using lens the multi-rays interfere at lens focus. ( ) ( ) [ ] ( )0 0 112242 21 ''''1 δωδδδ −−−−−− +++++= ++++= tiNiNii tNttt eEttererer EEEE   ( )0 02 2 ' '1 '1 δω δ δ − − − − − = ti i NiN eEtt er er ∞→ < Nand rIf 1' ( )0 02 '1 ' δω δ − − − = ti it eE er tt E In the case of zero absorption, no energy being taken out of the waves, using Stokes relations 2 1'&' rttrr −=−= ( )0 02 2 1 1 δω δ − − − − = ti it eE er r E 2 0 2 20 2 0 cos 4 : cos 2 : θ λ π δ θλ π δ nd nd = =
  • 75.
    75 InterferenceSOLO Multiple Beam Interferencefrom a Parallel Film ∞→ < Nand rIf 1 ( ) ti i i r eE er er E ω δ δ 02 1 1       − − = − − ( )0 02 2 1 1 δω δ − − − − = ti it eE er r E Let compute the Reflected and Transmitted Irradiances: ( ) ( ) ( ) ( ) 024 2 * 0022 * cos21 cos12 1 1 1 1 I rr r EE er er er er EEI i i i i rrr δ δ δ δ δ δ −+ − =      − − − − =∝ − − ( ) ( ) 024 22 * 002 2 2 2 * cos21 1 1 1 1 1 I rr r EE er r er r EEI iittt δδδ −+ − = − − − − =∝ − Using lens the multi-rays interfere at lens focus we found that in the case of zero absorption, no energy being taken out of the waves, using Stokes relations 2 1'&' rttrr −=−= 0III tr =+ ( ) ( )[ ] ( ) ( )[ ] ( ) 0222 2222/sin21cos 2/sin1/21 2/sin1/2 2 I rr rr Ir δ δδδ −+ − = −= ( ) ( )[ ] ( ) 0222 2/sin21cos 2/sin1/21 1 2 I rr It δ δδ −+ = −= We see that
  • 76.
    76 The transmission ofan 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 Multiple Beam Interference from a Parallel Film Return to Table of Content
  • 77.
    77 Gas RefrectometerSOLO S 1S 2 S 2T 1T f 1C 2 C D D 1C 2C E Rayleigh'sInterferometer 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 1 T 1 C 2 C D D 1C 2C E 1 G 2 G 1 2 Jamin's Interferometer t S 1 T E 1 M 2 G 1 2 Mach-Zehnder Interferometer 2M 3 M 4M 2T 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 3 1, 2 2 n nVN Tpng Reyleigh’s Interferometer Jamin’s Interferometer Return to Table of Content
  • 78.
    78 Interferometers History SOLO References 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/ Return to Table of Content
  • 79.
    January 5, 201579 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
  • 80.
    80 J.C. Wyant, “Introductionto Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
  • 81.
    81J.C. Wyant, “Introductionto Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
  • 82.
    82J.C. Wyant, “Introductionto Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
  • 83.
    83J.C. Wyant, “Introductionto Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
  • 84.
  • 85.
  • 86.
  • 87.
  • 88.
    88 Field and linearinterferometers 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 Interferometer Mach-Zehnder Interferometer Fabry-Perot Interferometer Sagnac Interferometer Gires-Tournois Etalon
  • 89.
    89 Field and linearinterferometers InterferenceSOLO Moire Interferometer (see Moire pattern) Holographic Interferometer Near-field Interferometer Fringes 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 http://en.wikipedia.org/wiki/List_of_types_of_interferometers Fizeau Interferometer Rayleigh Interferometer Twyman-Green Interferometer
  • 90.
    90 Intensity and nonlinearinterferometers InterferenceSOLO http://en.wikipedia.org/wiki/List_of_types_of_interferometers Intensity Interferometer Intensity Optical Correlator Frequency-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 Interferometer Ramsey Interferometer Mini Grail Interferometer Hanbury-Brown Twiss Interferometer
  • 91.
    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.
    92 http://www.grahamoptical.com/phase.html How Phase ShiftingWorks (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.
  • 93.

Editor's Notes

  • #6 M.V. Kleine &amp; T.E. Furtak, “Optics”, John Wiley &amp; Sons,2nd Ed., 1986, pg.26
  • #9 E. Hecht, “Optics”, Addison Wesley, 4th Ed., 2002, pp.406-407
  • #20 http://www.europhysicsnews.com/full/39/article5.pdf#search=&amp;apos;Humphrey%20Lloyd%20%26%20Conical&amp;apos;
  • #22 http://en.wikipedia.org/wiki/Wilhelm_Karl_Ritter_von_Haidinger E. Hecht, “Optics”, Addison Wesley, 4th Ed., 2002, pp.403-404
  • #24 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 306-312 http://de.wikipedia.org/wiki/Jules_C%C3%A9lestin_Jamin F.A. Jenkins, H.E. White, “Fundamentals of Optics”, McGraw Hill, 4th Ed., 1976, pp.282-284
  • #25 Eliott, pg. 49
  • #26 E. Hecht, “Optics”, Addison Wesley, 4th Ed., 2002, pp.404-405 J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
  • #27 E. Hecht, “Optics”, Addison Wesley, 4th Ed., 2002, pp.404-405 J.C. Wyant, “Introduction to Interferometric Optical Testing (SC213)”, Optical Science Center, University of Arizona, http://www.optics.arizona.edu
  • #28 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 289-290
  • #31 http://www.optics.arizona.edu/jcwyant/JoseDiaz/MichelsonInterferometer-BroadSource.htm http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html J.R. Meyer-Arendt,”Introduction to Classic and Modern Optics”, 3th Ed., Prentice Hall, 1989, pp.213
  • #32 http://www.optics.arizona.edu/jcwyant/JoseDiaz/MichelsonInterferometer-BroadSource.htm http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html J.R. Meyer-Arendt,”Introduction to Classic and Modern Optics”, 3th Ed., Prentice Hall, 1989, pp.213
  • #34 J.R. Meyer-Arendt,”Introduction to Classic and Modern Optics”, 3th Ed., Prentice Hall, 1989, pp.217 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1972, pp. 312-316
  • #35 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 310-311 http://de.wikipedia.org/wiki/Jules_C%C3%A9lestin_Jamin http://galaxy.eti.pg.gda.pl/katedry/kmoe/pracownicy/Pawel.Wierzba/otp/two_beams_interferometers.pdf
  • #36 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 269 F.G. Smith, J.H. Thompson, “Optics”, John Wiley &amp; Sons, 1971, pp. 258 F.A. Jenkins, H.E. White, “Fundamentals of Optics”, McGraw Hill, 4th Ed., 1976, pp.282-284
  • #37 J.R. Meyer-Arendt,”Introduction to Classic and Modern Optics”, 3th Ed., Prentice Hall, 1989, pp.223
  • #38 http://hal.archives-ouvertes.fr/docs/00/25/43/39/PDF/ajp-jp4199606C456.pdf
  • #39 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 303-306
  • #41 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 311-312 http://de.wikipedia.org/wiki/Jules_C%C3%A9lestin_Jamin http://www.metrology.pg.gda.pl/full/2008/M&amp;MS_2008_205.pdf
  • #42 http://en.wikipedia.org/wiki/Hanbury-Brown_and_Twiss_effect M. Françon, “Optical Interferometry”, Academic Press, 1966, p.181 http://www.powershow.com/view/cd86b-OGZhN/Hanbury_Brown_and_Twiss_Effect_powerpoint_ppt_presentation
  • #43 http://en.wikipedia.org/wiki/Gires-Tournois_etalon
  • #44 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #45 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #46 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #47 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #48 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #50 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 – http://homepage.univie.ac.at/Franz.Embacher/KinderUni2005/waves.gif
  • #51 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 – http://homepage.univie.ac.at/Franz.Embacher/KinderUni2005/waves.gif
  • #52 * Mathieu, J.P., “Optics – Part 1: Electromagnetic Optics”, Pergamon Press, 1975, pg. 21 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #53 E. Hecht, “Optics”, 4th Ed., Addison Wesley, 2002, p.398 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #54 E. Hecht, “Optics”, 4th Ed., Addison Wesley, 2002, p.398 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #55 E. Hecht, “Optics”, 4th Ed., Addison Wesley, 2002, p.398 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #56 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #58 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #59 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #60 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 – http://en.wikipedia.org/wiki/Lloyd’s
  • #61 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #62 Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.136-137 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 4th Ed., 1979, pp.91-93
  • #63 Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.136-137 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 4th Ed., 1979, pp.91-93
  • #64 M.V. Klein &amp; T. Furtak, “Optics”, 2nd Ed., John Wiley &amp; Sons, 1986, pp. 284-288
  • #65 M.V. Klein &amp; T. Furtak, “Optics”, 2nd Ed., John Wiley &amp; Sons, 1986, pp. 284-288
  • #66 M.V. Klein &amp; T. Furtak, “Optics”, 2nd Ed., John Wiley &amp; Sons, 1986, pp. 284-288
  • #67 M.V. Klein &amp; T. Furtak, “Optics”, 2nd Ed., John Wiley &amp; Sons, 1986, pp. 284-288
  • #68 M.V. Klein &amp; T. Furtak, “Optics”, 2nd Ed., John Wiley &amp; Sons, 1986, pp. 284-288
  • #69 http://en.wikipedia.org/wiki/Wilhelm_Karl_Ritter_von_Haidinger E. Hecht, “Optics”, Addison Wesley, 4th Ed., 2002, pp.403-404
  • #70 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #71 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 262 -
  • #72 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 323 - 329
  • #73 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 323 - 329
  • #74 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 323 - 329
  • #75 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 323 - 329
  • #76 Born &amp; Wolf, “Principles of Optics”, 5th Ed., Pergamon Press, 1975, pp. 323 - 329
  • #77 http://en.wikipedia.org/wiki/Fabry%E2%80%93P%C3%A9rot_interferometer
  • #78 F.A. Jenkins, H.E. White, “Fundamentals of Optics”, McGraw Hill, 4th Ed., 1976, pp.282-284 M. Kerker, “Scattering of Light and Other Electromagnetic Radiation”, Academic Press, 1969, pp. 31-39