Chapter 8 Optical Interferometry

Transcription

1 Chapter 8 Optical Interferometry Lecture Notes for Modern Optics based on Pedrotti & Pedrotti & Pedrotti Instructor: Nayer Eradat Spring 009 4/0/009 Optical Interferometry 1

2 Optical interferometry Interferometer is an instrument design to exploit the interference of light and fringe patternsthat that results from the optical path difference. Interferometers are extended to acoustic and radio waves as well. Here we will explore two kinds 1) Michelson MorleyInterferometer, Morley a two beam, amplitude divisiondevice. device. ) Fabry Perot interferometer a multiple beam, amplitude division device 4/0/009 Optical Interferometry

3 The Michelson interferometer (1881) Used in applications such as proving special theory of relativity, measuring hyperfine structure in line spectra, tidal effect of the moon on the earth. Principle i of operation is shown in the graph. In the graph of the interferometer the path difference between the beams traveling along the two perpendicular paths is: Δ = beam with respect to the optical axis. For normal beam θ = 0 Δ p = d = m d = m for m= 0,1,,3,... will form constructive interference and will repeat every / so long as d < L c separation stays smaller than the coherence length. p d cos θ (see the fig. b) where θ measures the inclination of the The device has two optical axis at right angle Movable mirror Compensator Beam splitter Fixed mirror 4/0/009 Optical Interferometry 3

4 Fringe analysis of Michelson interferometer Now the optical system is equivalent of the plane air film and the fringes of equal inclination are formed and can be viewed dby looking through hthe path h4 or using a telescope in that path. Fringes are made up of concentric circles centered around the optical axis with intensity of I=4I0 cos δ π where the phase difference is δ = kδ= Δ and Δ=Δ p +Δr Here we have a relative π phase shift between the two beams because the i reflection coefficient from two sides of the mirror differs by 1 = e π 1 Δ=Δ p +Δ r = dcosθ + = m+ dcos θ = m, m= 0,1,,... for dark fringes. d d The center fringe is dark. For the normal rays at the center d = m and m max = is order of central dark fringe and outer fringes have lower orders. We invert the fringe order for convenience d d by interoducing another integer p= mmax m= m p = d m p = d 1 cos θ where p=0,1,,3,.. dark fringes ( ) Now the central fring's order is zero. When optical path difference is decreasing the fringes move inward and disappear at the center. When optical path difference is increasing the fringes originate at the center and move outward. 4/0/009 Optical Interferometry 4

5 Fringe counting with Michelson interferometer Wavelength and distance measurement When optical path difference is decreasing the fringes move inward and disappear at the center. When optical path difference is increasing the fringes originate at the center and move outward A smaller mirror spaceing leads to an increase in angular separation Δθ for a given fringe interval. dcos θ = m, m= 0,1,,... for dark fringes. Δm d sin θ Δ θ = Δ m Δ θ = Differenciate θ with respect to m d sinθ this means for a smaller optical path difference the fringes are more widely separated. When d = / ( standing waves form ) then cos θ = m the entire field encompasses one fringe or m max =1. d For a mirror translation Δ d the number of fringes passing the center (cosθ = 1 and m= ) is Δ m = Δd in all of these senarios is constant and we are playing with finge number and d. This suggests an experimental method to measure small mirror movements with a known or measure when Δd is known. 4/0/009 Optical Interferometry 5

6 Applications of the Michelson interferometer 1) Measurement of thin film thickness ) Measurement of index of rfraction of a gass filled in a cell of length L placed in one arm of the interferometer. The fringe count Δm is done as the gas is evacuated fromt the cell. The optical path difference is: Δ d = nl- L= L( n 1) = Δm n= Δ m+ 1 L 3) Measurement of temperature of a gas or index change as a function of temperature. You need ( ) to know T = f n for the mate rial to use it. 4) Determination of the wavelength difference between two closely spaced spectral lines and ' or ν and ν' used in the lab to measure the difference between the wavelengths of the sodium yellow lines. Two kinds of fringes can be observed with iththe Michelson interferometer. t 1) when the mirrors are absolutely perpendicular to each other virtual fringes of equal inclination are observed. ) When the mirors are not quite perpendicular ( small deviation ), an air wedge appears between the mirrors and fringes of equal thickness may form localized at the mirrors that appear as straight lines parallel to the intersection of the two mirrors. For large deviations the fringes appear as hyperbolic curves. 4/0/009 Optical Interferometry 6

7 Variations of the Michelson InterferometerTwyman Green Testing of a prism and lens for variations in their index of refraction or surface imperfections. Point source with a collimating lens Fringes of equal thickness (parallel lines) appear that show imperfections of optical systems Fringes boost interferometer resolution A recent article on increasing resolution of the measurements by interferometry 4/0/009 Optical Interferometry 7

8 Fringes of equal thickness in neighborhood of a candle by Michelson Interferometer 4/0/009 Optical Interferometry 8

9 Mach Zehnder Interferometer Used in aerodynamic research. It uses two beamand and amplitude splitting. Advantage over Michelson interferometer is possibility of making fringes at the object of study so that both can be photographed together. Test Chamber Identical Chamber 4/0/009 Optical Interferometry 9

10 The Fabry Perot Interferometer The fabry-perot interferometer is composed of two semi-transparent parallel plates. Very simple in structure but a very precise measurement tool. The interference pattern is composed of superposition of the multiple beams of transmitted and reflected light. Applications: precision wavelength measurements, hyperfine spectral structures,.measuring refractive indices of gasses, calibration of the standard meter in terms of wavelength. The interferometer is composed of the inner surfaces of the cavity mirrors that usually polished to /50 and coated with silver or aluminium layer ~50nm. Outer surfaces are cut at small angle so that reflection from them does not interfere. When the mirror spacing is fixed the cavity is called etalon. The beam from source S generates multiple cohernt beams in the interferometer the emerging rays are brought together at P. The path difference between the succesive beams is Δ = ndcos θ, for air n = 1 p f i f Δ r = π or no effect on the interference. The condition for bright fringes is dcosθ = m i The fringe pattern, concentric rings, are the fringes of equal inclination. Cavity 4/0/009 Optical Interferometry 10

11 Different arrangements of the Fabry Perot interferometer a) Extended source and a fixed plate spacing. Circular fringe pattern. b) Point source and a variable plate spacing. A detector records intensity as a function of the plate spacing. With a laser sources the lenses are not needed. 4/0/009 Optical Interferometry 11

12 Fabry Perot transmission: the Airy function Goal: calculating the irradiance transmitted through a Fabry-Perot interferometer. Two identical mirrors separated by a distance d that have real reflection and transmission coefficients r, t and no absorption so r + t = 1. Cavity roundtrip-time τ = d/ V = nd/ c. We want to express the transmitted electric field ET in terms of incident electric field EI. Propagation factor is the ratio of a taveling monochromatic plane wave with electric field E zt, to E z, t P F pg g ( ) ( 0 0 ) E( z0 +Δ z, t0 +Δt) = For the traveling plane monochromatic wave with no change in optical media E( z0, t0) i ω( t0+δt) k( z0+δz) E( z0 +Δ z, t0 +Δt ) E 0e i ( ωδt-kδz ) ( Δ, Δ t ) = = = e i E( z, t ( ωt0 kz0) ) Ee P z t F 0 0 Right-going electric field incident on cavity from the left E ( ) = ( ) Amplitude time-dependent of intercavity right-going electric field E1 E01 t e At time t + τ the right-going intercavity field is sum of two parts E + 1 ( t+ τ) = tei ( t+ τ) + r PF ( Δ z = d, Δ t = τ) E + 1 ( t) Incident field transmitted through mirror 1 The right-going intercavity field that existed at mirror one cavity round-trip time τ earlier iω( t+ τ) iω( t+ τ) iωt i( ωτ-kd) ( + τ ) = + ( ) E t e te e r E t e e I 01 After some time the intercavity field will settle to a ( τ ) ( ) I = E e 0I iωt ( + ) ( + ) steady-state value then E t+ = E t = E and E e = te e + r E e iω t τ iω t τ + iωt i -kd I 01 e ωτ + t π E01 = E 0I where δ = kd = d is the round-trip phase shift. i 1 re δ iωt ( ) 4/0/009 Optical Interferometry 1

13 Fabry Perot transmission: the Airy function Goal: calculating the irradiance transmitted through a Fabry-Perot interferometer. + t E01 = E 0I where δ = kd is the round-trip phase shift. iδ 1 re iω( t+ τ /) + ( + τ /) = = ( Δ =, Δ = τ /) ( ) ET t E0Te tpf z d t E1 t = te + 01 Portion of E ( t) that is transmitted through the mirror after propagating the length of the cavity in half round-trip time or τ / te te E t e E E 1 re 1 re iδ / iδ / iωt T ( + τ /) = iδ 0I 0T = E iδ 0I Irradiance I T E T E T Transmitance T * 0 0 ( 1 r ) iδ / * + iδ / ( E0It e )( E0It e ) i δ + i δ ( 1 re )( 1 re ) I E E = = = I E E E E * T 0T 0T * * I 0I 0I 0I 0I 4 t T = I r r cosδ = 1+ r r cosδ = ( / /) E e i ωτ δ + iωt 01 ( Transmittance of a parallel plate from chapter 7) 1 δ The Airy functoin T = where we used cosδ = 1 sin 4r δ 1+ sin 1 r ( ) T 4/0/009 Optical Interferometry 13

14 Coefficient of Finesse 4r Fabry called the coefficient of finesse. whith that the Airy function can be expresses as ( ) The Airy functoin: F r = ( 1 r ) 1+ F sin ( δ ) ( F) ( δ ) 1 Tmax = 1 for sin / = 0 T = δ + F Tmin = 1/ 1 + for sin / =± 1 Coefficient of finesse is a very sensitive function of the reflection coefficient r. For 0 < r < 1 0 < F < ( ) ( + ) I I T T 1 1/ 1+ F I T 1/ 1 F T,max T,min max min Fringe contrast = = = = F the larger the fringe contrast the better the cavity. Fringe contrast = F = T,min min Coefficient of finesse Note the difference between fringe visibility, IT,max IT,min and fringe contrast IT,min Regardless of r, Tmax = 1 is at δ = m( π) and Tmin = 1/ ( 1 + F) at δ = ( m+ 1/ ) π and limt = 0 is never zero but appoaches it r 1 min as r 1. The fringes become very sharp compared to the Michelson interferometer. 4/0/009 Optical Interferometry 14

15 Finesse π F πr 4r 4F Finesse of a cavity: F= = is different than the coerfficient of finesse: F = = 1 r 1 π ( r ) separation between the transmitance peaks FSR Goal: showing that the finesse F= = Full width at half maximum of the peaks FWHM 1 1 Transmittance of a cavity T = = 1+ F sin ( δ / ) 1+ ( 4 F / π ) sin ( δ / ) Free spectral range FSR of a cavity: the phase separation between the adjacent transmittance peaks. ( ) ( ) δfsr = δm+ 1 δm = m+ 1π πm= π Half width at half maximum HWHM δ of the transmittance peak can be found by setting: ( ) 1/ 1 1 4F δ δ π T = = sin 1 sin = = 1+ ( 4 F / π ) sin ( δ /) π 4F Phase of a maximum is mπ and half-maximum just after of a maximum has a phase of δ = mπ + δ 1/ δ1/ δ1/ δ1/ π sin mπ + sin = = F For the narrow transmission peaks δ is small. This approximation is good for cavities with high reflectivity mirrors the finesse is high and the transmittance peaks are narrow. δ 1/ 1/ 1/ δ fsr π π = π / F F = δ = δ = FWHM Thus finesse of a cavity is the ratio of FSR and FWHM of the cavity transmittance peaks. 4/0/009 Optical Interferometry 15 1/

16 Finesse as a figure of merit 4r 4 Coerfficient of finesse: F = = F The figure of merit finesse: ( 1 r ) π π F πr δ π F= δ fsr = = = 1 r FWHM The transmittance is function of round-trip phase shift δ which in-turn is a function of d, Mirror spacing (cavity length) ν, Frequency One of these are varied in different modes of operation. n, Index of refraction of the inter-mirror material As a result FWHM and FSR change by varying the independent variable. But their ratio, finesse of the cavity, stays constant. That is why finesse is a good figure of merrit to identify a cavity. Finesse only depends on the reflectivity of the mirrors ( δ π ) 1/ limf = 0 r 0 limf = r 1 FSR = for different modes of operation corresponds to variations in different parameters: d fsr fsr is the free spectral range of a variable length Fabry-Perot interferometer. ν is the free spectral range of a variable input frequency Fabry-Perot interferometer. fsr c = ν fsr is the free spectral range of a variable input wavelength Fabry-Perot interferometer. 4/0/009 Optical Interferometry 16

17 Scanning Fabry Perot Interferometer The cavity length can be changed by moving one of the mirrors of a Fabry-Perot interferometer. Thiscanbeusedtomeasurewavelengthof of a monochromatic source. At maximum transmission T max the phase difference between the interfering waves is: π δ = kd = d = mπ m = 0, ± 1, ±,... Round-trip phase gain dm = m and d fsr = dm+ 1 dm = For more accurate measurements this is not good because the best actuators (piezoelectric) have have ~ 10 nm accuracy. 4/0/009 Optical Interferometry 17

18 Scanning Fabry Perot Interferometer We can send two wery closely spaced wavelengths and simultaneously into a Fabry-Perot cavity. Nominal length of the cavity d = 5cm 1 Nominal wavelength of the cavity =500 nm is very close to the and and by slight adjustment of the cavity length we get the FSR of both wavelengths. We have to make sure both transmission peaks correspond to the same mode order m then we can write: 1 = d1/ m Δ Δd 1 =Δ = ( d d1) = Δd = = d / m m ( d1/ 1) 1 d It is hard to know absoloute wavelength and length of the cavity but we can replace the 1 and d1 with the nominal values and d and conclude: Δ Δd = d The lengths we are detecting The lengths we are measuring 1 4/0/009 Optical Interferometry 18

19 Resolving power of a Fabry Perot cavity The minimum wavelength difference that can be determined by a cavity is limitted by the width of the 1/ min Δ min transmittance peaks of the two s. Minimum resolvable between the cavity lengths, Δ d = FWHM of the peaks. Resolving criteria is Δd Δd Δd δ fsr kd fsr d fsr d fsr F = = = Δ d 1/ = = δ1/ kδd1/ Δd1/ F F Δ Δd = Δmin Δdmin Δd1/ d = = d d Δ dmin = Δd 1/ Δmin = df df R= = = mfresolving power of a FP cavity Δ min d Where m = is the mode number associated with the nominal and d of the cavity. Large resolving power corrsponds to: 1) higher modes that means large mirror spacing d m can't change much π r ) large finesse that is achieved by r as close to 1 as possible F = 1 r 6 Good Fabry -Perot interferometers have resolving powers as high as 10 about order of magnitude better than prisms and gratings. = ( ) 4/0/009 Optical Interferometry 19 min

20 Resolving power of a Fabry Perot cavity (maximum) The maximum wavelength difference that can be determined by a cavity Δmax is limitted by the overlap of the mth order transmittance peak of and m + 1th order transmittance peak of. 1 1 Δ Δ d = m m = m between 1 and d fsr = ( m+ 1 ) m = between 1 and 1 Difference in cavity length associated with adjacent paeks has to be equal to the FSR of the variable length interferometer Δ 1 Δ d = d fsr m = Δ / m Δ / m max max 1 = = Δmin / ( mf) Δ max F = Another good dfeature of the finesse: high hfinesse cavities ii have ability to resolve very Δ min small wavelength differncescan over a large wavelength range. The Δ Δ = max max fsr is also known as wavelength FSR: F 4/0/009 Optical Interferometry 0

21 4/0/009 Optical Interferometry 1

22 4/0/009 Optical Interferometry

23 4/0/009 Optical Interferometry 3

24 4/0/009 Optical Interferometry 4

25 4/0/009 Optical Interferometry 5