Multiple Beam Interference

Transcription

1 MutipeBeamInterference.nb James C. Wyant 1 Mutipe Beam Interference 1. Airy's Formua We wi first derive Airy's formua for the case of no absorption. ü 1.1 Basic refectance and transmittance Refected ight E r E i Ir + t t' r'  d + º + t t' r' H2 p-3l  Hp-1L d M Transmitted ight E t E i t t' I1 + r' 2 Âd + r' 4  2 d + º + r' 2 Hp-1L  Hp-1L d M Phase due to OPD d 2 p o 2 n d Cos@qD

2 2 James C. Wyant MutipeBeamInterference.nb ü 1.2 Stokes Reations If there are no osses a wave's propagation must be reversibe. Equations derived from Stokes Reations t t' + r 2 1 t r' + r t 0 t t' 1 - r 2 r -r' Therefore R + T 1; t t' T; R r 2 r' 2 ü 1.3 Refected Light E r E i Ir + t t' r' Âd I1 + r' 2  d + º + r' 2 Hp-2L  Hp-2L d MM p-2 E r E i r + t t' r'  d r' 2 n  n d n0 E r E i r + t t' r'  d 1 - r' 2 Hp-1L  Hp-1L d 1 - r' 2  d Substitute r -r' and et pø. E r E i d t t' r  r r 2  d E r E i r 1 - r2  d - t t'  d 1 - r 2 Âd 1 - Ir2 + t t'm  d E r E i r 1 - r 2  d E r E i R I1 -  d M 1 - R  d R H2-2 Cos@dDL I r E r E * r 1 + R 2-2 R Cos@dD But 1 - Cos@dD 2 Sin@dê2D 2

3 MutipeBeamInterference.nb James C. Wyant 3 I r 4 R Sin@d ê 2D 2 H1 - RL R Sin@d ê 2D 2 ü 1.4 Transmitted Light E t E i t t' I1 + r' 2  d + r' 4  2 d + º + r' 2 Hp-1L  Hp-1L d M E t E i t t' Let pø. E t E i E t E i t t' 1 - r' 2  d T 1 - R  d 1 - r'2 p  p d T r' 2  d 1 + R 2-2 R Cos@dD T 2 H1 - RL R Sin@d ê 2D 2 ü 1.5 Comments on Airy's Formua I r and are known as Airy' s formua. Note that if there are no osses so T + R 1, Let F, the coefficient of finesse, be given by F Then I r 4 R H1 - RL 2 F Sin@d ê 2D F Sin@d ê 2D F Sin@d ê 2D 2 This shows again that the refected and transmitted ight are compementary. I r does go to zero for a vaues of R, but it goes to 1 ony in the imit that R Ø 1. goes to 1 for a vaues of R, but it goes to zero ony in the imit that R Ø 1. As RØ1 the transmitted ight becomes narrow bright fringes on a dark background. As RØ1 the refected ight becomes narrow dark fringes on a bright background. I r + 1.

4 4 James C. Wyant MutipeBeamInterference.nb ü 1.6 Pots of Transmission and Refection 1.0 Transmission Fringes Refectivity 0.04 HGreenL, 0.18 HRedL, 0.8 HBueL 0.8 Irradiance Order Number Refection Fringes Refectivity 0.04 HGreenL, 0.18 HRedL, 0.8 HBueL 0.8 Irradiance For a maximum Sin@dê2D 2 0. d 2 2 p n d Cos@qD m p o or for a maximum in the transmitted ight 2 n d Cos@qD m Order Number Going through the same procedure for refected ight we find that for a minimum in the refected ight 2 n d Cos@qD m This is the same resut found previousy for two beam interference.

5 MutipeBeamInterference.nb James C. Wyant 5 For zero refection a beams after the first refection are subtracting from the first refection. ü 1.7 Low Refectivity Approximations If R is sma, F is aso sma, and the equations for the refected and transmitted ight can be approximated as I r º F Sin@d ê 2D 2 F 2 H1 - Cos@dDL º 1 - F Sin@d ê 2D F H1 - Cos@dDL 2 These equations are characteristic of two-beam interference. ü 1.8 Fringe Sharpness Sharpness of fringes convenienty measured by their haf intensity width which for transmitted ight is the width between points on either side of maximum where intensity has faen to haf its maximum vaue. finesse Hseparation of adjacent fringesl ê Hwidth of haf maxl For intensity at haf max d 2 m p ± e 2 Thus F SinA e 4 E2 If F is very arge e is sma so SinB e 4 F2 º K e 4 O 2 finesse 2 p e 2 p 4 í F finesse p 2 F p R H1 - RL

6 6 James C. Wyant MutipeBeamInterference.nb 2. Absorbing Coatings If the two surfaces of the pate are identica, but there are osses, the vaue of can be determined as foows. d 2 p o 2 n d Cos@qD + 2 f f is the phase change upon refection for each surface. We can sti write or T 2 H1 - RL R Sin@d ê 2D 2 T 2 H1 - RL F Sin@d ê 2D 2 However, now we have osses of an amount A so we must write R + T + A 1 or T H1 - RL - A It foows that 1 - A 1 - R T max 1 - A 1 - R F Sin@d ê 2D A T 2 The effect of absorption is to reduce transmitted intensity and shift fringes. For the maximum transmitted intensity the important quantity is A. Even though A may be very sma, if T is aso sma (R arge), A may become arge and the T T maximum transmitted intensity may be very sma. As an exampe et R 99.7% and A 0.2%, so T is approximatey 0.1%. T max is now 11%. However, et R 99.7% and A 0.29%. Now T is 0.01% and T max becomes 0.11%. What is happening physicay is that whie for each refection there is very itte oss, the refectivity is so high that there are many effective refections and the tota oss becomes arge. Whie the phase shift due to f is normay not a probem at norma incidence there may be a probem at non-norma incidence because f is a function of poarization. At norma incidence f is an equivaent to an increase of f o /2p in optica thickness of the pate. The refection case is more compicated because first refection experiences no absorption. As a resut the interference pattern does not go to zero.

7 MutipeBeamInterference.nb James C. Wyant 7 3. Fabry-Perot The mutipe beam interference fringes from two highy refecting surfaces iuminated near norma incidence are used in the cassica Fabry-Perot interferometer. A Fabry-Perot interferometer is usefu for spectroscopy. Narrow bright circuar fringes are obtained. For a bright fringe of order m m d 2 p 2 n d Cos@qD + f o p

8 8 James C. Wyant ü Transmission Fringes MutipeBeamInterference.nb

9 MutipeBeamInterference.nb ü Refection Fringes James C. Wyant 9

10 10 James C. Wyant MutipeBeamInterference.nb ü 3.1 Resoving power If more than one waveength is present we see a superposition of the transmission pattern for each waveength. Let there be two waveengths present, 1 and D. Our criterion for resoution is that the ines are just resovabe if the haf maximum intensity of the peak of order m for one waveength coincides with the haf maximum intensity of the peak of order m for the second waveength The eft side of the figure shows the individua intensity contours of two Fabry-Perot fringes that are just resoved. The right side shows the two intensity contours added to give the observed effect. It shoud be noted that some books such as Born & Wof have a different criterion. They choose a separation such that the sum of the two intensities equas that of the maximum. This agrees with the Rayeigh criterion that if we had a sinc 2 function the intensity maximum of one ine woud coincide with the minimum of the second ine. Using this criterion we woud get a resoution equa to 0.97 the resoution we get using our criterion. The phase difference between the two interfering beams is d 4 p n d Cos@qD + 2 f where 2f is much smaer than 4 p Dd 1ê2 2 p finesse Dd - 1 d D n d Cos@qD. D d Dd 2 p m 2 p finesse m HfinesseL resoving power near norma incidence m º 2 n d D m HfinesseL m p resoving power º 2 n d finesse Thus the resoution is proportiona to the mirror separation. 2 F m p R H1 - RL As an exampe et the finesse be 30 (R º 0.9), nd 4 mm, 500 nm, then the resoving power is º 5 x 10 5 and D is

11 MutipeBeamInterference.nb James C. Wyant 11 As an exampe et the finesse be 30 (R 0.9), nd 4 mm, 500 nm, then the resoving power is 5 x and is nm. The question is why not keep increasing the resoution by increasing the separation of the Fabry-Perot pates indefinitey? The probem is that we woud have an overapping of orders. The waveength difference at which overapping takes pace is caed the free spectra range. ü 3.2 Free Spectra Range Overapping takes pace when order m of waveength D fas on top of order m+1 of waveength 1. Hm + 1L 1 m 2 m H 1 + DL Thus D m D FSR m 2 Near norma incidence D FSR 2 2 n d Cos@qD 2 n d ; Dn FSR c 2 n d So increasing the resoving power by increasing the cavity thickness gives a reduction in the FSR. D FSR m ; D res D FSR finesse p D res 2 F m HfinesseL p R 1 - R ü 3.3 Spectrometry with a Fabry-Perot Etaon The prism does gross separation to eiminate, or at east reduce, the FSR probem. The Fabry-Perot gives high resoution. The foowing figure shows some actua interference fringes obtained using a Fabry-Perot etaon with a prism spectrometer (Ref: Born & Wof).

12 12 James C. Wyant MutipeBeamInterference.nb ü 3.4 Scanning Fabry-Perot The scanning Fabry-Perot is usefu when ony a few discrete waveengths are present as is often the case with a aser. The scanning can be achieved by mounting one of the Fabry-Perot mirrors on a PZT. Since for a given fringe 2 d m ; 2 D d m D 2 d D Thus Dd d D As d is varied different waveengths wi be transmitted through the Fabry-Perot and the oscioscope dispay wi show the waveengths present.

13 MutipeBeamInterference.nb James C. Wyant 13 ü 3.5 Spherica Fabry-Perot The figure beow shows one form of a spherica Fabry-Perot. In the drawing the ower haf of each spherica mirror is totay refecting and the upper haf is semi-transparent. The center of curvature of each mirror is ocated on the opposite mirror. In the paraxia region the path difference between the initia ray I J and ray I J I' J' I J is equa to 4 d, where d is the distance between M 1 and M 2. Instead of obtaining a series of parae emerging rays originating from a singe incident ray, as is the case with a pane parae pate Fabry-Perot, we have a series of overapping rays traveing aong JT. The phase difference between consecutive rays is given by f 2 p H4 dl which is independent of the incination of the rays and their azimuth within the imits of the Gaussian approximation. The intensity expression is the same as for a reguar parae pate Fabry-Perot, except we have a fat tint a over the fied. If rays are incined to the axis third-order aberrations produce variations of the path difference and we find the fat tint surrounded by circuar fringes. We can reduce aberrations by pacing two identica circuar diaphragms centered on M 1 and M 2. This interferometer is we suited to the arge path differences corresponding to a high spectra resoution. 4. FECO (Fringes of Equa Chromatic Order) Previousy we were concerned with mutipe-beam fringes produced by monochromatic radiation. In some cases it is better to use a white ight source. In this section we wi combine a mutipe beam interferometer with a spectrometer to measure thickness variations. For transmission I max where d 2 p 2 n d Cos@qD + 2 f 1 + F Sin@d ê 2D 2 o f is the phase change on refection at each surface. A schematic diagram of a FECO interferometer is shown beow. Both the sampe and the reference surface must have high refectivity so high finesse mutipe beam interference fringes are obtained. The sampe is imaged onto the entrance sit of a spectrometer

14 14 James C. Wyant MutipeBeamInterference.nb If n 1 and q 0 for a bright fringe of order m f p + 2 d m It shoud be noted that for a given fringe d constant and m 2 d m - f p Soving for the height difference across a sampe is compicated since f f[]. However, with many coatings f can be considered to be independent of over the sma spectra region used for the anaysis. (For more detais see Born & Wof or Jean Bennett, JOSA 54, p. 612 (1964). The foowing drawing shows two fringes in the FECO output. The goa is to find the surface height difference between points 1 and 2. d m - f p 2 and d 2 - d 1 m - f p For point 1 and fringe orders m and m + 1 2,m - 1,m 2 m - f p 1,m m f p 1,m+1 Thus, m - f p 1,m+1 1,m - 1,m+1 and d 2 - d 1 1,m+1 2,m - 1,m 1,m - 1,m+1 2

15 MutipeBeamInterference.nb James C. Wyant 15 The foowing figure shows some actua FECO interference fringes (Ref: Born & Wof). Since d 2 - d 1 is proportiona to 2,m - 1,m, the profie of the cross-section of an unknown surface is obtained by potting a singe fringe on a scae proportiona to the waveength. The spectroscopic sit is in effect seecting a narrow section of the interference system and each fringe is a profie of the variation of d in that section since there is exact point-to-point correspondence between the seected region and its image on the sit. Sma changes in d are determined by measuring sma changes in. There are no ambiguities as to whether a region is a hi or a vaey. There are no ambiguities at a discontinuity as we woud have with monochromatic ight where it is difficut to determine which order beongs to each fringe. Surface height variations in the Angstrom range can be determined. Two disadvantages are 1) we are getting data ony aong a ine and 2) the sampe being measured must have a high refectivity.