OPTICS: the science of light 2 nd year Physics FHS A2 P. Ewart

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1 OPTICS: the science f light nd ear Phsics FHS A P. wart Intrductin and structure f the curse. The stud f light has been an imprtant part f science frm its beginning. The ancient Greeks and, prir t the Middle Ages, Islamic schlars prvided imprtant insights. With the cming f the Scientific Revlutin in the 6 th and 7 th centuries, ptics, in the shape f telescpes and micrscpes, prvided the means t stud the universe frm the ver distant t the ver small. Newtn intrduced a scientific stud als f the nature f light itself. Tda Optics remains a ke element f mdern science, nt nl as an enabling technlg, but in Quantum Optics, as a means f testing ur fundamental understanding f Quantum Ther and the nature f realit itself. Gemetrical ptics, studied in the first ear, ignred the wave nature f light and s, in this curse, we fcus particularl n Phsical Optics where the primar characteristic f waves viz. interference, is the dminant theme. It is interference that causes diffractin the bending f light arund bstacles. S we begin with a brief résumé f elementar diffractin effects befre presenting, in chapter, the basics f scalar diffractin ther. B using scalar ther we ignre the vectr nature f the electric field f the wave, but we return t this aspect at the end f the curse when we think abut plarizatin f light. Scalar diffractin ther allws us t treat mathematicall the prpagatin f light and the effects f bstructins r restrictive apertures in its path. We then intrduce a ver pwerful mathematical tl, the Furier transfrm and shw hw this can help in slving difficult diffractin prblems. Furier methds are used ver widel in phsics and recgnise the inter-relatin f variables in different dimensins such as time and frequenc r space and spatial frequenc. The latter cncept will be useful t us in understanding the frmatin f images in ptical sstems. Having established the mathematical basis fr describing light we turn t methds f analsing the spectral cntent f light. The spectrum f light is the primar link between ptics and atmic phsics and ther sciences such as astrphsics. The basis fr almst all instruments fr spectral analsis is, again, interference. The familiar Yung s slit, tw-beam, interference effect, in which the interference arises frm divisin f the wave-frnt b tw slits, is generalised t multiple slits in the diffractin grating spectrmeter. The alternative methd f prducing interference, b divisin f amplitude, is then cnsidered. Again we begin with the case f tw beams: the Michelsn interfermeter and mve n t multiple-beam interference in the Fabr-Pert interfermeter. These devices are imprtant tls and pla a ke rle in mdern laser phsics and quantum ptics. The reflectin and transmissin f light at bundaries between dielectric media is an imprtant feature f almst all ptical instruments and s we then cnsider hw the phsics f wave reflectin at bundaries can be engineered t prduce surfaces with high r partial reflectivit r even n reflectivit at all. Finall we return t the vectr nature f the electric field in the light wave. The directin in which the -field pints defines the plarizatin and this can be randm in un-plarized light, fixed in space in linearl plarized light r rtating in space in the case f ellipticall r circularl plarized light. We will stud hw t prduce, manipulate and analse the state f plarizatin f light.

2 . Waves and Diffractin. Mathematical descriptin f a wave Figure. u u cs( kzt ) t : phase change with time, kz : phase change with distance, k = : arbitrar initial phase r u u e e i i( kzt ) Nte: this cnventin fr a wave travelling frm left t right i.e. in the psitive z directin fllws that used in Quantum Mechanics t describe wave functins.. Interference Additin f amplitudes frm tw surces gives interference e.g. Yung's slits: r r P d dsin D Figure. Yung's slits Tw slits separated b d illuminated b mnchrmatic plane waves Amplitude u p at a pint P a large distance, D, frm the slits u i( kr t ) u i( kr t ) u p e e (.) r r Putting ( r r ) d sin, r r r, intensit is: u I p 4 cs ( kdsin ) (.) r.3 Phasrs The amplitude f a wave is represented b the length f a vectr n an Argand diagram. The phase f the wave is represented b the angle f the vectr relative t the

3 Imaginar Real axis f the Argand diagram. i The phasr is then: ue u Real Figure.3 Phasr diagram xample: Yung's slits. u /r Figure.4 Phasr diagram fr tw slit prblem. u p u /r Amplitude frm each slit n screen: u r Phase difference, wing t path difference d sin : kdsin Resultant amplitude is then u u p cs( / ) r The intensit is therefre: I p u 4 cs ( kdsin ) (.3) r.4 Diffractin frm a finite slit Mnchrmatic plane wave incident n aperture f width a. Observatin plane at large distance D frm aperture. Amplitude in plane f aperture: u per unit length. +a/ d r + sin r P -a/ sin D Figure.5 Cntributins t amplitude at P frm elements d in slit.

4 An infinitesimal element f length d at psitin cntributes at P an amplitude: ud ( ) e i r The phase factr ( ) k( r sin ). The ttal amplitude at P arising frm all cntributins acrss the aperture: u p a / u ikr ik sin. e e d r (.4) The intensit is then: I p I(0) sinc a / where ka sin sinc () Figure.6 Intensit pattern frm single slit, I p = I(0)sinc. The first minimum is at, asin Hence angular width f diffractin peak is: (.6) a.5 Diffractin frm a finite slit: phasr treatment +a/ r P -a/ asin D Figure.7 Cnstructin shwing elements at extreme edges f aperture cntributing first and last phasrs

5 On axis, the phasr elements sum t R p Off axis, 0 successive phase shifts between adjacent phasrs bend the phasr sum t frm a sectin f a regular plgn. / R P Figure.8 (a) Phasr diagram fr finite slit shwing resultant R p fr and 0 The phase difference between first and last phasrs fr is R P kasin In the limit as the phasr elements 0 the phasrs frm an arc f a circle f radius R. The length f the arc is R and the length f the chrd representing the resultant is R p. R R P Figure.8 (b) Phasr diagram in the limit as phasr elements 0. The amplitude at relative t the amplitude at = 0 : length f chrd Rsin( / ) sinc( / ) length f arc R. Then the intensit at : I() = I(0) sinc (/) = I(0) sinc Phasr arc t first minimum Phasr arc t secnd minimum Figure.9 Phasr diagram shwing mimina fr increasing phase shift between extremes f slit as increases. The first minimum ccurs when the phasr arc bends t becme a full circle i.e. the phase difference between first and last phasr elements the angular width is: a

6 .6 Diffractin in dimensins Recall that the amplitude resulting frm a plane wave illuminatin f an aperture f the frm f a slit f width a in the - directin (eqn.4): u p u r e a / ikr a / e ik sin. Cnsider the aperture t have a width b in the x -directin, then the angular variatin f the diffracted amplitude in the x -directin is : u In - D we have : u p u e r p b / ikr e b / e b / ikr ik sin. x a / b / a / dx d u( x, ). e ik(sin. xsin. ) dxd x z Figure.0 General -D aperture in x, plane. u(x,) is the amplitude distributin functin fr the aperture. Fr a circular aperture f diameter a the diffractin pattern is a circular Bessel Functin. The angular width t the first minimum is:. Intensit a x Figure.. Pint Spread Functin fr circular aperture. A pint surce imaged b a lens f fcal length f and diameter a gives a pattern with a minimum f radius r = f/a. This is the Pint Spread Functin r instrument functin analgus t the impulse functin f an electrical circuit giving its respnse t a -functin impulse.

7 . Fraunhfer Diffractin S far we have cnsidered diffractin b (a) Apertures r slits illuminated b plane waves (b) Observatin at a large distance where the phase difference between cntributins frm secndar surces in the diffracting plane separated b is given t a gd apprximatin b: k sin. These are special cases where the phase difference is a linear functin f the psitin in the diffracting aperture.. Fraunhfer diffractin Definitin: A diffractin pattern fr which the phase f the light at the bservatin pint is a linear functin f psitin fr all pints in the diffracting aperture is Fraunhfer Diffractin. B linear we mean that the wave frnt deviates frm a plane wave b less than / 0 acrss the diffracting aperture. R a a R surce R R bserving pint diffracting aperture Figure. Wavefrnts incident n and exiting frm a plane aperture. Alternativel, fr ( R ) R a R 0a / Fraunhfer diffractin is the diffractin bserved in the image plane f an ptical sstem.

8 P O A B C Figure. Fraunhfer cnditin fr plane waves: image is at infinit as surce is at fcal length frm lens. Cnsider a pint surce at the fcal pint f a lens s that cllimated light (plane waves) are incident n an aperture behind the lens. The image f the surce is at. Fraunhfer Diffractin hwever will be bserved at P if BC O P u v P O quivalent lens sstem Figure.3 Fraunhfer diffractin bserved in the image plane f a lens. If the bservatin pint P lies in the image plane f the lens s that curved wavefrnts cnverge frm the lens t P then n plane waves are invlved. The lens and diffracting aperture hwever can be replaced b an equivalent sstem where diffractin f plane waves ccurs. Nte hwever that this means plane waves are nt necessar t bserve Fraunhfer diffractin. The ke criterin is that... the phase varies linearl with psitin in the diffracting aperture. A further cnsequence f nting that Fraunhfer diffractin is bserved in the image plane is that the psitin f the aperture is nt imprtant.

9 (a) O P (b) O P Figure.4 quivalent lens sstem shwing that Fraunhfer diffractin is independent f psitin f aperture. Diffractin and wave prpagatin Cnsider a plane wave surface at -z. This reprduces itself at a secnd plane z = 0. Hugens secndar surces in the wave frnt radiate t a pint P in the secnd plane. ds n r P -z 0 z Plane wave surface Figure.5 Hugens secndar surces n plane wave at -z cntribute t wave at P. The amplitude at P is the resultant f all cntributins frm the plane at z. uds ikr u p (n,r) e r u is the amplitude frm element f area ds. (n,r) is the bliquit factr - this accunts fr the fact that the wave prpagates nl in the frward directin. n is a unit vectr nrmal t the wave frnt and is a prprtinalit cnstant - t be determined. We determine b a self-cnsistenc argument i.e. the plane wave at -z must reprduce itself at z = 0. We cnsider the amplitude at a pint P a distance q frm the wave such that q = m where m is an integer and m>>. i.e. P is a large distance frm O, a pint n the wave frnt ling n a nrmal thrugh P. We nw cnstruct elements f the wavefrnt f equal area A centred n O.

10 n r n q P The first element is a circle, the n th is an annulus f uter radius Figure.6 Cnstructin f elements f equal area n plane wavefrnt. ( n n ) A Cnsequentl the difference in distance r frm successive elements t P is cnstant A r rn rn q Therefre the phase difference between waves frm successive elements is als cnstant: A q Hence we ma treat cntributins frm each element f the wavefrnt as a Phasr. [Nte: we ignre, fr the mment, the small difference in amplitude at P between successive elements arising frm the small increase in distance r n as n increases. We als ignre the small change in (n,r) ] Add cntributins f elements (phasrs) until the last phasr added is ut f phase with the first. The area f the wavefrnt cvered b these elements is the First Half Perid Zne, st HPZ. n Figure.7 Cnstructin f the First Half Perid Zne

11 The difference in path-length frm the uter element f the st HPZ t P and frm O t P is /. / q O q Figure.8 Phase shift f arises at edge f st HPZ P The radius f the st HPZ is is given b: q Recalling ur diffractin integral we write the cntributin t the amplitude at P frm the st HPZ: u u q Frm the phasr diagram, the amplitude frm the st HPZ is the length f the phasr arc, u. The resultant R is then the diameter f the circle f which the phasr arc defines half the circumference: R u The resultant phasr lies alng the imaginar axis s: R iu Add further elements until the final phasr is in phase with the first i.e. a phase difference f. The area f the wavefrnt nw defines the first Full Perid Zne st FPZ. The resultant frm the st FPZ is nt exactl zer wing t the term /r (inverse square law fr intensit) and the bliquit factr (n,r). Adding further elements gives a slw spiral. Figure.9 As n resultant f znes tends t half the resultant f the st HPZ

12 Adding cntributins frm the whle wave (integrating ver infinite surface) gives resultant equal t ½ the st HPZ. Therefre R iu Self-cnsistenc demands that this wave at P matches the riginal wave at O: Hence u p u iu i i u ds (n,r) r ikr e (.) This is the Fresnel-Kirchff diffractin integral.

13 3. Furier methds in Optics 3. The Fresnel-Kirchff integral as a Furier Transfrm The Fresnel-Kirchff diffractin integral tells us hw t calculate the field U p in an bservatin plane using the amplitude distributin u in sme initial plane i u ds u ( ) r ikr p n, r e, (3.) where the limits f integratin will be defined b the bundar f the aperture. We simplif b: Ignring the bliquit factr i.e. put ( n, r), Restricting t ne dimensin: ds dx, Ignring the r term b cnsidering nl a small range f r, Using the Fraunhfer cnditin: ikr ikr ik sin x e e e, ikr Absrbing e int the cnstant f prprtinalit: Since the integral will be zer wherever the amplitude functin u(x) is zer the limits f integratin can be safel extended t infinit. The amplitude u p as a functin f angle is then: where k sin. u p ix A( ) u( x) e dx, (3.) We nte that A( is the Furier transfrm f u(x). The Fraunhfer diffractin pattern is prprtinal t the Furier transfrm f the transmissin functin (amplitude functin) f the diffracting aperture. Mre precisel: the Fraunhfer diffractin pattern expressed as the amplitude, as a functin f angle, is the Furier transfrm f the functin representing the amplitude f the incident wave, as a functin f psitin in the diffracting aperture. The Fraunhfer diffractin is expressed as a functin f k sin where is the angle f the diffracted wave relative t the wave vectr k f the wave incident n the aperture. The inverse transfrm relatin is: i ( ) x u( x) A e d (3.3)

14 3. The Cnvlutin Therem The cnvlutin f tw functins f(x) and g(x) is a new functin, h(x), defined b: h( x) f ( x) g( x) f ( x) g( x x) dx (3.4) The Furier transfrm, F.T., f f (x) is F () The Furier transfrm, F.T., f g (x) is G () The Furier transfrm, F.T., f h (x) is H () The Cnvlutin Therem states that the Furier transfrm f a cnvlutin f tw functins is the prduct f the Furier transfrms f each f the tw functins: H( ) F( ). G( ) (3.5) 3.3 Sme useful Furier transfrms and cnvlutins (a) We can represent a wave f cnstant frequenc as a functin f time t. v( t) V e it F. T.{ v( t)} V( ) V ( ) i.e. V( represents the spectrum f a mnchrmatic wave f frequenc and is a delta functin in frequenc space. Figure 3. A wave f cnstant frequenc (mnchrmatic) and its Furier transfrm Alternativel the inverse transfrm relatins allw us t represent the F.T. f a delta functin: v( t) V ( t t ) as inverse F. T.{ v( t)} V( ) V e it

15 (b) The duble slit functin, i.e. tw delta-functins separated b d : (c) A cmb f delta functins: v ( x) ( x d ) V ( ) cs( d ) b b V(x) x S x V( ) The F.T. f v c (x) is: Figure 3. A cmb f -functins and their transfrm N vc ( x) m 0 ( x mx ) i sin( Nxs ) Vc ( ) e sin( xs ) where ( N ) x s The factr e i is simpl the cnsequence f starting ur cmb at x = 0. This factr can be eliminated b shifting ur cmb t sit smmetricall abut the rigin. This result illustrates the Shift Therem. s (d) The tp-hat functin: a vd ( x) fr x Vd ( ) a sinc( a) v ( x) 0 fr d x a [What wuld be the result if the tp-hat was shifted t sit between x = 0 and x = a?]

16 Nw sme useful cnvlutins: (e) The duble slit: ) ( ) ( ) ( x v x v x v d b s (f) The grating functin: ) ( ) ( ) ( x v x v x v d c g (g) The triangle functin: ) ( ) ( ) ( x v x v x v d d This is a self-cnvlutin. The self-cnvlutin is knwn als as the autcrrelatin functin. 3.4 Furier Analsis A peridic functin V(t) ma be represented b a Furier series. ) sin( ) cs( ) ( t p s t p c c t V p p p p (3.6) V(t) is the result f snthesis f the set f Furier cmpnents. Furier analsis is the reverse prcess - finding the cmpnents (amplitude and phase) that make up V(t). The cefficients are fund b integrating the functin ver a perid f the scillatin. t t V c t t p t V c t t p t V s p p d ) ( d ) ) cs( ( d ) )sin( ( In general: (3.8) d ) ( (3.7) ) ( 0 t e t V A e A t V t ip p t ip p p This last expressin represents a Furier transfrm - suggesting that this peratin analses the functin V(t) t find the amplitudes f the Furier cmpnents A p.

17 3.5 Spatial frequencies Cnsider a plane wave falling nrmall n an infinite screen with amplitude transmissin functin: u( x) sin( x) i.e. a grating with peridic pattern f width s d This defines the spatial frequenc: s d The Fraunhfer diffractin pattern is then: where k sin. We find: A( ) s u( x) e i.e. sin 0 r d The sinusidal grating has a Fraunhfer diffractin pattern cnsisting f zer rder and + first rders d /. / s ix dx A( ) 0 except fr 0, An additinal spatial frequenc n will lead t additinal first rders at n /. [Nte: a finite screen will result in each rder being spread b the diffractin pattern f the finite aperture, i.e. the spread functin f the aperture.] 3.6 Abbé ther f imaging We cnsider an bject cnsisting f an infinite screen having a sinusidal transmissin described b a functin u(x) s that the amplitude transmissin repeats with a spacing d. This acts as an bject at a distance u frm a lens f fcal length f. d u(x) u f Furier plane Figure 3.3 Object u(x) imaged b lens t v(x). Diffractin rders are waves with parallel wave vectrs at angles and +d. A lens brings these parallel waves t a fcus as pints in the fcal plane separated b a = f/d. Apart frm a phase factr, the amplitude in the fcal plane is the F.T. f u (x). This plane is the Furier plane. Zer and first rder pints act as cherent surces giving tw-beam interference at psitins bend the fcal plane. In the image plane, distance v frm the lens, the a v D s d v(x)

18 interference pattern is maximall sharp, v = f + D. The interference pattern is a sinusidal fringe sstem with spacing: D d a Frm gemetr d d u v Hence: u v f Fr a finite grating the pints will be spread b diffractin at the effective aperture f the grating. [Nte that we can describe such a grating as a cnvlutin f an infinite sine wave with a tp-hat functin.] An bject amplitude distributin ma be snthesised b a set f sinusidal functins. ach Furier cmpnent with a specific spatial frequenc cntributes + rders t the diffractin pattern at specific angles t the axis. The aperture a f the lens and bject distance u determine the maximum angle max frm which light ma be cllected. Diffractin rders at angles greater than max d nt cntribute t the final image. The crrespnding spatial frequencies will be missing frm the image. Higher spatial frequencies cntribute t sharp edges in the bject distributin. The lack f high spatial frequencies in the image leads t blurring and lss f reslutin. [Nte: the discussin s far is valid nl fr cherent light i.e. light waves having a fixed phase relatinship acrss the aperture in the bject plane. In practice fr micrscpic bjects this cnditin is partiall fulfilled even fr white light illuminatin.] 3.7 The Cmpund Micrscpe Figure 3.4 shws the arrangement f the cmpund micrscpe. Basicall a ver shrt fcal length lens, the bjective, frms a real, inverted, image f the specimen in the image plane, giving a linear magnificatin f v/u. The ee-piece is basicall a simple magnifier used t view the real image which is lcated at the fcal length f the eepiece giving a virtual image at infinit. This allws viewing with minimum eestrain. The minimum dimensin f spatial structure in the bject d min that can be reslved is such that the assciated diffractin rder will be at the maximum angle max that can be cllected b the bjective lens. sin max d Spatial frequencies, having dimensins smaller than d min, will diffract t larger angles, miss the bjective, and thus nt appear in the image. The minimum spatial dimensin d min min

19 Figure 3.4 The Cmpund Micrscpe. The bject at distance u frm bjective with fcal length f O is imaged at distance v This real image is at the fcal length f frm the eepiece giving an angular magnificatin / where / is the angle subtended b the real image if it was at the near pint f the ee, distance D. Apprximatel, u and v L, the length f the tube. In this apprximatin the magnificatin is f O M DL f O f that can be reslved ma be increased b immersing the bjective and bject in il f refractive index n ; the il immersin bjective: n sin max d n sin 0 max is the Numerical Aperture and defines the ultimate reslutin f the device. min max Figure 3.5. First rder diffracted waves frm spatial structures < d min are cllected b the lens and interfere in the image plane with zer rder waves t frm sinusidal structure in the image. Light frm smaller spatial structures (higher spatial frequencies) are diffracted t angles > max, miss the bjective and d nt interfere with zer rder in the image. 3.7 Diffractin effects n image brightness Nrmal image brightness is determined b the f/n. f the ptical sstem i.e. f/d A where d A is the limiting aperture. When the image size appraches the rder f the PSF ~ /d A light is lst frm the image b diffractin. This is diffractin limited imaging. Fr nn-diffractin limited imaging: Fr diffractin limited imaging: Image brightness Image brightness d A 4 d A

20 8 4 Optical instruments and fringe lcalizatin Optical instruments fr spectrscp use interference t prduce a wavelength-dependent pattern. The interfering beams are prduced either b divisin f wavefrnt r b divisin f amplitude. The diffractin grating divides the wavefrnt int multiple beams. The Michelsn divides the amplitude int tw beams and the Fabr-Pert interfermeter divides the amplitude int multiple beams. It is imprtant t knw where t lk fr the fringes. Befre lking at specific instruments we cnsider the general questin f fringe lcalizatin. 4. Divisin f wavefrnt (a) Tw-slit interference, Yung's Slits nn-lcalised fringes Figure 4. Yung's slit fringes are bserved thrughut the regin bend the screen cntaining the tw slits. The fringes are nn-lcalized and usuall bserved under the Fraunhfer cnditin. (b) N-slit diffractin, the diffractin grating. t Figure 4. Diffractin grating fringes. f Again we usuall bserve the Fraunhfer cnditin. A mnchrmatic plane wave is diffracted i.e. suffers cnstructive interference at angle. Parallel light interferes at infinit r in the fcal plane f a lens. The fringes are lcalized at infinit r in the image plane f the instrument.

21 4. Divisin f amplitude The interference ma invlve tw beams (Michelsn) r multiple beams (Fabr-Pert). The situatins are mdelled b reflectin f light frm a surce at tw surfaces. The surce ma be a pint r extended and the surfaces ma be at an angle (wedged) r parallel. The images f the surce in the reflecting surfaces act as tw effective surces. 4.. Pint surce (a) Wedge. P P O Figure 4.3 A pint surce O prvides images P, P' in reflecting surfaces frming a wedge. This sstem is equivalent t -pint surces r Yung's slit situatin. Therefre the fringes are nn-lcalized fringes f equal thickness.. (b) Parallel P P O Figure 4.4 A pint surce reflected in tw parallel surfaces again prvides tw images P, P' This is similar t the wedge situatin with -pint surces. The fringes are nn-lcalized fringes f equal inclinatin.

22 4.. xtended surce (a) Wedge R R P P O S Figure 4.5. xtended surce OS prvides tw images PR and P'R' b reflectin at wedged reflecting surfaces. ach pint n the extended surce prduces nn-lcalized fringes. Overlap f all these patterns gives n visible fringes. Hwever at the apex f the wedge the path difference is zer and is the same fr all pints n the effective surces s fringes are visible in this regin. The zer rder fringe is a straight line fringe in the plane f the wedge. Other lw-rder fringes ma be seen if the surce is nt t large and the wedge angle nt t big. The fringes are f equal thickness and lcalized in the plane f the wedge e.g. Newtn's Rings. (b) Parallel images t surce t=x path difference xcs t=x Figure 4.6 Upper figure shws tw images f extended surce b reflectin in parallel slab f thickness t. Lwer figure shws fringes f equal inclinatin frmed in fcal plane f a lens b light frm the tw images f the surce. Clse t plate verlapping patterns lead t n visible fringes. At large distance the fringes becme wider and exceed the displacement f the verlap. Fringes becme visible and are fringes f equal inclinatin and lcalized at infinit. These fringes are mre cnvenientl bserved in the fcal plane f a lens. e.g. the ee. Reflecting surfaces separated b t lead t tw images separated b t r x = t. Parallel light at an angle f inclinatin t the axis frm equivalent pints n the effective circular fringe cnstant

23 surces are brught tgether in the fcal plane. The path difference is xcs and the phase difference : x cs (4.) Bright fringes (cnstructive interference) ccurs when the phase difference p (p = integer) r xcs p (4.) Fr small angles the angular size f the fringes is given b p p x Hence radii f fringes in fcal plane f lens with fcal length f : f rp rp (4.3) x As x increases, fringes get clser tgether. As x decreases 0 fringes get larger and fill the field f view. The behaviur f the fringes frmed b parallel surfaces will be imprtant fr the Michelsn and Fabr-Pert interfermeters.

24 5 The diffractin grating spectrgraph 5. Interference pattern frm a diffractin grating Cnsider a plane wave f wavelength incident nrmall n a reflecting r transmitting grating f N slits separated b d. The amplitude cntributed b each slit is u and the intensit f the interference pattern is fund b adding amplitudes and taking the squared mdulus f the resultant. () N = I 4 N = 0 Figure 5. Intensit pattern and assciated phasr diagram fr -slit interference I( ) 4u cs ( ) (5.) where d sin Principal maxima at 0, n, f intensit 4u. One minimum beween principal maxima. () N = 3 N = 3 9 I 4 0 Figure 5. Phasr diagrams fr 3-slit interference and intensit pattern Using phasrs t find resultant amplitude (a) 0, n Principal maxima f intensit 9u. Tw minima between principal maxima. (b) / 3 Minimum / zer intensit (c) Subsidiar maxima f intensit u (d) 4 / 3 Minimum / zer intensit

25 (3) N = 4 Principal maxima at 0, n f intensit 6u. Three minima between principal maxima. In general we have principal maxima at 0, n, intensit N. (N ) minima at n and width f principal maxima. N N N I 4 0 N N mn Figure 5.3 Phasr diagrams fr N-slit interference and intensit pattern Amplitude f N phasrs: Hence intensit: A u ue ue... ue i i i( N) N sin ( ) I( ) I(0) (5.) sin ( ) 5. ffect f finite slit width Grating f N slits f width a separated b d is a cnvlutin f a cmb f N functins f(x) with a single slit (tp-hat functin) g(x): N a f ( x) ( x pd) ; g( x), fr { x p0 h( x) f ( x) g( x) Using the Cnvlutin Therem with, a }; g( x) 0, fr { x a } F( ) F. T.{ f ( x)}, G( ) F. T.{ g( x)} and H( ) F. T.{ h( x)} H( ) F( ). G( )

26 Hence H( ) where kdsin and kasin 5.3 Diffractin grating perfrmance sin ( ) sin ( ) I( ) I(0). (5.3) sin ( N ) ( ) ) 5.3. The diffractin grating equatin The equatin fr I ( ) gives the psitins f principal maxima, 0, n, n is an integer: the rder f diffractin (this is als the number f wavelengths in the path difference). Fr nrmal incidence n the grating. Principal maxima ccur fr d sin n (5.4) 5.3. Angular dispersin The angular separatin d between spectral cmpnents differing in wavelength b d d n (5.5) d d cs Reslving pwer Figure 5.4 (a)the phase shift min the change in between the maxima and first minimum in phase space, crrespnds t an angular separatin min in real space. (b)the fringes are reslved if the angular width t the first minimum min equals the angular separatin f the tw wavelengths. Principal maxima fr wavelength ccur fr a phase difference f n. The change in phase difference between the maximum and the first minimum is min and min (5.6) N d sin

27 Angular width t first minimum min is fund frm d d cs d Thus the phase difference between the maximum and first minimum is: min d cs. min N min (5.7) Nd cs The angular separatin f principal maxima fr and is fund frm (5.5): d n d d cs The reslutin criterin is: Hence the Reslving Pwer is: n d cs min nn (5.8) Free Spectral Range The n th rder f and (n + ) th rder f ( FSR ) lie at same angle. { n d sin ( n )( ) }. Hence verlap ccurs fr these wavelengths at this angle. The Free Spectral Range is thus: FSR (5.9) ( n ) Nte: the Reslving Pwer n and the FSR / n. 5.4 Blazed (reflectin) gratings The Blaze angle is set t reflect light int the same directin as the diffracted rder f chice fr a given wavelength. Fr incident angle and diffracted angle the blaze angle will be : ( ) where and satisf the grating equatin d(sin sin) n (5.0)

28 Reflected light Diffracted light (a) Reflected light Diffracted light (b) Figure 5.5 (a) Diffractin angle Reflectin angle fr rdinar grating. (b) Blazed grating reflects light at same angle as diffracted rder (a) Order (b) 0 (c) Order Figure 5.6 (a) Grating intensit pattern and single slit diffractin pattern. (b) ffect f single slit diffractin envelpe n grating diffractin intensit fr unblazed grating. (c) Grating intensit pattern fr blaze set t reflect light int nd rder.

29 5.5 ffect f slit width n reslutin and illuminatin Cnsider the imaging frming sstem cnsisting f tw lenses f fcal length f and f. The image f a slit f width x s has a width: f xi x s f (5.) f f x s (a) f grating x s f (b) Figure 5.7 (a) Image frming sstem t image slit f widthx s t image x i. (b) Images f slit are spectrall dispersed b diffractin at grating. Slit is imaged at angle frm diffractin grating leading t freshrtening b cs. In a diffractin grating spectrgraph the image is viewed at the diffractin angle and s is freshrtened b cs. f xi x s f cs The minimum reslvable wavelength difference, R, has an angular width R : n R R d cs Wavelengths having difference R are separated in the image plane f lens f b x R : where we used R nn Reslutin is achieved prvided: f x R (5.) Nd cs x x and the limiting slit width x s is then: i R f f xs r x s (5.3) Nd W

30 Nte: the ptimum slit width is such that the diffractin pattern f the slit just fills the grating aperture, W = Nd. x x s x R s x R : reslutin reduced b verlap f images at different wavelengths : reslutin nt imprved bend diffractin limit but brightness is reduced. The pint f this exercise is t cmpare the size f the slit image in the real spectrgraph with the size predicted using the theretical reslving pwer. The ke difference between the tw is that the image in the real spectrgraph, because it is bserved at the diffractin angle, is freshrtened b a factr cs. S this means that we have t make the bserved image size x i larger b a factr / cs in rder t cmpare like with like. The theretical reslving pwer leads us t a value fr the angular separatin f wavelengths that differ b R. Using ur expressin fr the angular dispersin we find this angular separatin t be: n R R d cs This is equatin (5.3). We nw need t find the spatial extent f the image, x R, crrespnding t this theretical reslutin and this is given simpl b the relatin fwhere the fcal length in this case is f then using, R nn : This is equatin (5.4). n xr f R d cs f x R Nd cs NB. This calculated theretical image size des nt include an freshrtening effect. Nw cnsider the size f the image in the real spectrgraph that will exhibit freshrtening. The size f the image frmed in the lens imaging sstem f the spectrgraph is determined b the linear magnificatin f the lens sstem: M = v/u = f /f. S the size f an image is related t the entrance slit width x s b the relatin: x f i x s f cs, i.e. When an image is viewed in the diffracted beam at angle there is a freshrtening b a factr f x i x i cs. Cnversel an image f size x i viewed in the diffracted beam crrespnds t an unfreshrtened image that will be larger b the same factr i.e. x i fxs f cs cs. This is equatin (5.). It is this image size withut the effect f freshrtening that needs t be cmpared t the theretical image size. In rder t achieve the theretical reslving pwer we need t have: In which case: Or the limiting slit width is fxs f f cs Nd cs x s f Nd x x We nte that if the slit is less than this limiting value the reslutin is nt imprved the image simpl becmes less bright we have diffractin limited imaging. i R

31 6 The Michelsn (Furier Transfrm) Interfermeter A tw-beam interference device in which the interfering beams are prduced b divisin f amplitude at a 50:50 beam splitter. M M / M t CP BS Detectr Light surce Figure 6. The Michlesn interfermeter. The beam splitter BS sends light t mirrrs M and M in tw arms differing in length b t. M is image f M in M resulting effectivel in a pair f parallel reflecting surfaces illuminated b an extended surce as in figure 5.6. CP is a cmpensating plate t ensure beams traverse equal thickness f glass in bth arms. 6. Michelsn Interfermeter Distance frm beam splitter t mirrrs differs b t in the tw paths, and is the angle f interfering beams t the axis. Resulting phase difference between beams: t cs xcs (6.) Cnstructive interference ccurs at p, where p is an integer, x cs = p. Thus n axis the rder f interference is p = x/. Smmetr gives circular fringes abut axis. The fringes are f equal inclinatin and lcalized at infinit. The are viewed therefre in the fcal plane f a lens. Fringe f rder p has radius r p in the fcal plane f a lens (fcal length, f, see sectin 5..(b). Tw-beam interference pattern: where, the wavenumber. r f rp (6.) x p I( x) I(0) cs I( x) I(0) cs x (6.3)

32 Input spectrum Detectr signal Interfergram Figure 6. Input spectrum f mnchrmatic surce and resulting interfergram btained frm scanning Michelsn interfermeter. 6. Reslving Pwer f the Michelsn Spectrmeter. Cnsider that we wish t reslve tw wavelengths and that differ b. The crrespnding wavenumbers are and and the prvide tw independent interfergrams s the resultant is the sum f the tw: I( x) I0( )[ cs x] I0( )[ cs x] Let the tw cmpnents have equal intensit: s I0( ) I0( ) I0( ) is the intensit f each interfergram at x = 0. Then I( x) I0( ) cs xcs x (6.4) x (a) I( ) x (b) I( ) x x max (c) I( ) x Figure 6.3 (a) Interfergram f surce cmpnent (b) interfergram f surce cmpnent. (c) Interfergram f cmbined light shwing added intensities (a) and (b). Nte visibilit f fringes ccles t zer and back t unit fr equal intesit cmpnents. T reslve the cmplete ccle requires a path difference x max

33 This lks like an interfergram f a light surce with mean wavenumber multiplied b an envelpe functin x. This envelpe functin ges cs first t a zer when a peak f interfergram fr first cincides with a zer in the interfergram fr. The visibilit (r cntrast) f the fringes ccles t zer and back t unit; the tell-tale sign f the presence f the tw wavelength cmpnents. The number f fringes in the range cvering the ccle is determined b the wavenumber difference. The instrument will have the pwer t reslve these tw wavenumbers (wavelengths) if the maximum path difference available, x max, is just sufficient t recrd this ccle in the envelpe f the interfergram. The minimum wavenumber difference min that can be reslved is fund frm the value f x max giving the ccle in the csine envelpe functin: min x max min (6.5) xmax This minimum reslvable wavenumber difference is the instrument width as it represents the width f the spectrum prduced b the instrument fr a mnchrmatic wave. Inst (6.6) xmax Hence the Reslving Pwer RP is: xmax RP. (6.7) 6.3 The Furier Transfrm spectrmeter Inst In Figure 6. we see that the interfergram lks like the Furier transfrm f the intensit spectrum. The interfergram prduced using light f tw wavenumbers and is I( x) I0( )[ cs x] I0( )[ cs x] In the case f multiple discrete wavelengths: I( x) I0( i )[ cs ix] I0( i ) I0( i )cs ix i First term n r.h.s. is ½ I where I is the ttal intensit at x = 0 and the Secnd term is a sum f individual interfergrams. Replacing cmpnents with discrete wavenumbers b a cntinuus spectral distributin, I(x) becmes: I ( x) I0 S( )cs x. d 0 i Inst i

34 where S ( ) is the pwer spectrum f the surce. Nw S ( ) 0 fr 0, s secnd term ma be written: F ( x) S( )cs x. d (6.8) S F(x) is the csine Furier Transfrm f F. T.{ S( )} I( x) I0 Or S( ) F. T. I( x) I0 (6.9) Apart frm a cnstant f prprtinalit the Furier transfrm f the interfergram ields the Intensit r Pwer Spectrum f the surce. See Figure 6.. The Michelsn interfermeter effectivel cmpares a wavetrain with a delaed replica f itself. The maximum path difference that the device can intrduce, x max, is therefre the limit n the length f the wavetrain that can be sampled. The lnger the length measured the lwer the uncertaint in the value f the wavenumber btained frm the Furier transfrm. Distance x and wavenumber are Furier pairs r cnjugate variables.[see equatin (7.9)] This explains wh the limit n the uncertaint f wavenumber (r wavelength) measurement Inst is just the inverse f x max. In essence this explains the general rule fr all interfermeters including diffractin grating instruments that: (6.0) Maximum path difference between interfering beams Inst 6.4 The Wiener-Khinchine Therem Nte: this tpic is NOT n the sllabus but is included here as an interesting theretical digressin. The recrded intensit I(x) is the prduct f tw fields, (t) and its delaed replica (t + ) integrated ver man ccles. (The dela x/c. ) The interfergram as a functin f the dela ma be written: ( ) ( t) ( t ) dt Taking the integral frm t be (6.) t we define the Autcrrelatin Functin f the field ( ) ( t) ( t ) dt (6.) The Autcrrelatin Therem states that if a functin (t) has a Furier Transfrm F(

35 F. T.{ ( )} F( ) F ( ). F( ) (6.3) Nte the similarit between the Autcrrelatin therem and the Cnvlutin Therem. The phsical analgue f the Autcrrelatin therem is the Wiener-Kinchine Therem. The Furier Transfrm f the autcrrelatin f a signal is the spectral pwer densit f the signal The Michelsn interfergram is just the autcrrelatin f the light wave (signal). Nte that and are related b a factr c where c is the speed f light. 6.5 Fringe visibilit Fringe visibilit and relative intensities Figure 6.3 shws an interfergram made up f tw independent surces f different wavelengths. The cntrast in individual fringes f the pattern varies and we define the visibilit f the fringes b I max I min V (6.4) I I max The fringe visibilit cmes and ges peridicall as the tw patterns get int and ut f step. The example shwn cnsisted f tw surces f equal intensit. The visibilit varies between and 0. If hwever the tw cmpnents had different intensit I ( ) and I ( ) then the envelpe functin f the interfergram des nt g t zer. The cntrast f the fringes varies frm I ( ) + I ( ) at zer path difference (r time dela) t a minimum value f I ( ) - I ( ). Denting the intensities simpl b I and I. min V max I I 0 I I 0 ; V min I I I I Hence V V min max I I I I which leads t: I I V V min max (6.5) min / V / V max Measuring the rati f the minimum t maximum fringe visibilit V min / V max allws the rati f the tw intensities t be determined Fringe visibilit, cherence and crrelatin

36 When the surce cntains a cntinuus distributin f wavelengths/wavenumbers the visibilit decreases t zer with increasing path difference x and never recvers. The tw parts f each f the Furier cmpnents (individual frequencies) in each arm f the interfermeter are in phase at zer path difference (zer time dela). At large path differences there will be a cntinuus distributin f interfergrams with a range f phase differences that average t zer and n stead state fringes are visible. The path difference x intrduced that brings the visibilit t zer is a measure f the wavenumber difference L acrss the width f the spectrum f the surce. L (6.6) x is the spectral linewidth f the surce. L A surce having a finite spectral linewidth i.e. ever light surce (!) ma be thught f as emitting wavetrains f a finite average length. When these wavetrains are split in the Michelsn, and recmbined after a dela, interference will ccur nl if sme parts f the wavetrains verlap. Once the path difference x exceeds the average length f wavetrains n further interference is pssible. The tw parts f the divided wavetrain are n lnger cherent. The Michelsn interfergram thus gives us a measure f the degree f cherence in the surce. A perfectl mnchrmatic surce (if it existed!) wuld give an infinitel lng wavetrain and the visibilit wuld be unit fr all values f x. The tw parts f the divided wavetrain in this case remain perfectl crrelated after an dela is intrduced. If the wavetrain has randm jumps in phase separated in time n average b sa c then when the tw parts are recmbined after a dela d < c nl part f the wavetrains will still be crrelated. The wavetrains frm the surce sta crrelated with a delaed replica nl fr the time c which is knwn as the cherence time. Thus we see that the Michelsn interfergram prvides us with the autcrrelatin functin r selfcrrelatin alng the length f the electrmagnetic wave emitted b the surce. (see sectin 4.3) In ther wrds the Michelsn prvides a measure f the lngitudinal cherence f the surce. [Nte. Light surces ma als be characterised b their transverse cherence. This is a measure f the degree f phase crrelatin the waves exhibit in a plane transverse t the directin f prpagatin. Mnchrmatic light emanating frm a pint surce will give spherical wavefrnts i.e. ever pint n a sphere centred n the surce will have the same phase. Similarl a plane wave is defined as a wave riginating effectivel frm a pint surce at infinit. Such a surce will prvide Yung s Slit interference n matter hw large the separatin f the slits. (The fringe width, f curse, will get ver tin fr large separatins.) If the slits are illuminated b tw separate pint surces, with the same mnchrmatic wavelength but with a small displacement, then tw sets f independent fringes are prduced. The displacement f the surces gives a displacement f the tw patterns. Fr small slit separatin this ma be insignificant and fringes will be visible. When, hwever, the slit separatin is increased the peaks f ne pattern verlap the trughs f the ther pattern and unifrm illuminatin results. The separatin f the slits in this case therefre indicates the extent f the spatial crrelatin in the phase f the tw mnchrmatic surces i.e. this measures the extent f the transverse cherence in the light frm the extended surce.]

37 7. The Fabr-Pert interfermeter This instrument uses multiple beam interference b divisin f amplitude. Figure 7. shws a beam frm a pint n an extended surce incident n tw reflecting surfaces separated b a distance d. Nte that this distance is the ptical distance i.e. the prduct f refractive index n and phsical length. Fr cnvenience we will mit n frm the equatins that fllw but it needs t be included when the space between the reflectrs is nt a vacuum. An instrument with a fixed d is called an etaln. Multiple beams are generated b partial reflectin at each surface resulting in a set f parallel beams having a relative phase shift intrduced b the extra path dcs between successive reflectins which depends n the angle f the beams relative t the axis. (See sectin 4.. (b)). Interference therefre ccurs at infinit the fringes are f equal inclinatin and lcalized at infinit. In practice a lens is used and the fringes bserved in the fcal plane where the appear as a pattern f cncentric circular rings. 7. The Fabr-Pert interference pattern This is dne in all the text bks (cnsult fr details). The basic idea is as fllws: d tr 6 e -i3 tr 4 tr tr 5 tr 3 tr tr 4 e -i tr e -i t t Figure 7. Multiple beam interference f beams reflected and transmitted b parallel surfaces with amplitude reflectin and transmissin cefficients r i, t i respectivel. Amplitude reflectin and transmissin cefficients fr the surfaces are r, t and r, t, respectivel. The phase difference between successive beams is: d cs (7.) An incident wave e -it is transmitted as a sum f waves with amplitude and phase given b: t it i( t ) i( t ) 0tt e 0tt rr e 0tt r r e... etc. Taking the sum f this Gemetric Prgressin in r r e i

38 it t 0tt e i rr e and multipling b the cmplex cnjugate t find the transmitted Intensit: It tt 0t t r r r r cs writing 0 I0, r r R and tt T, and cs ( sin / ) : T I t I0 (7.) 4R ( R) sin / ( R) If there is n absrptin in the reflecting surfaces T ( R) then defining 4R ( R) (7.3) I t I 0 sin / (7.4) This is knwn as the Air Functin. See figure 7. I( ) m (m+) Figure 7. The Air functin shwing fringes f rder m, m+ as functin f. 7. Observing Fabr-Pert fringes The Air functin describes the shape f the interference fringes. Figure 7. shws the intensit as a functin f phase shift. The fringes ccur each time is a multiple f. d cs m (7.5) m is an integer, the rder f the fringe. The fringes f the Air pattern ma be bserved b a sstem t var d,, r A sstem fr viewing man whle fringes is shwn in

39 figure 7.3. An extended surce f mnchrmatic light is used with a lens t frm the fringes n a screen. Light frm an pint n the surce passes thrugh the F.P. at a range f angles illuminating a number f fringes. The fringe pattern is frmed in the fcal plane f the lens. Fabr-Pert interfermeter Lens Screen d xtended Surce Lens Figure 7.3. Schematic diagram f arrangement t view Fabr-Pert fringes. Parallel light frm the Fabr-Pert is fcussed n the screen. Frm equatin (7.5) the m th fringe is at an angle m m cs m (7.6) d The angular separatin f the m th and (m + ) th fringe is m is small s m m+ = Fr small angles cs cs leads t: m m d cnstant m Therefre the fringes get clser tgether twards the utside f the pattern. The radius f the fringe at m is m ( ) f m f cs (7.7) d An alternative methd t view fringes is Centre spt scanning. A pint surce r cllimated beam ma be used as the surce and imaged n a pinhle. Light transmitted thrugh the pinhle is mnitred as a functin f d r Fringes are prduced f rder m linearl prprtinal t d r, (/. This als has the advantage that all the available light is put int the detected fringe n axis. 7.3 Finesse The separatin f the fringes is in -space, and the width f each fringe is defined b the half-intensit pint f the Air functin i.e. I / I / t 0 when sin /

40 The value f at this half-intensit pint is sin / differs frm an integer multiple f b a small angle s we have: / The full width at half maximum FWHM is then 4 The sharpness f the fringes ma be defined as the rati f the separatin f fringes t the halfwidth FWHM and is dented b the Finesse F r F (7.8) R F (7.9) ( R) S the sharpness f the fringes is determined b the reflectivit f the mirrr surfaces. [Nte: 3 F. checks that the quadratic equatin fr R has been slved crrectl!] ~ ( R) 7.4 The Instrument width The width f a fringe frmed in mnchrmatic light is the instrumental width: Inst (7.0) F Inst is the instrument width in terms f the apparent spread in wavenumber prduced b the instrument fr mnchrmatic light. Fr n-axis fringes (cs: d d d d Hence d Inst d Inst and : F Inst (7.) df

41 7.5 Free Spectral Range, FSR Figure 7.4 shws tw successive rders fr light having different wavenumbers, and th ( ). Orders are separated b a change in f The ( m ) rder f wavenumber ma verlap the m th rder f ( ) i.e. changing the wavenumber b mves a fringe t the psitin f the next rder f the riginal wavenumber. d ( m ) d and ( )d m This wavenumber span is called the Free Spectral Range, FSR: FSR (7.) d I( d) m th (m+) th d Figure 7.4 Fabr-Pert fringes fr wavenumber and ( ) bserved in centrespt scanning mde. The m th -rder fringe f and ( ) appear at a slightl different values f the interfermeter spacing d. When the wavenumber difference increases s that the m th rder fringe f ( ) verlaps the (m+) th rder f the wavenumber difference equals the Free Spectral Range, FSR In figure 7.4 the different rders fr each wavelength (wavenumber) are made visible b changing the plate separatin d. (Because changing d will change ). The phase can be varied b changing d, r. See equatin (7.5). In figure 7.3 the different rders fr a given wavelength are made visible b the range f values f. If the surce emits different wavelengths, fringes f the same rder will appear with different radius n the screen. 7.6 Reslving Pwer The instrumental width ma nw be expressed as: FSR Inst (7.3) F df

42 Tw mnchrmatic spectral lines differing in wavenumber b R are just reslved if their fringes are separated b the instrumental width: R Inst Figure 7.5 Reslutin criterin: light f tw wavenumbers, R is reslved when the separatin f fringes fr and R is equal t the instrument width Inst. As in Figure 7.4 the fringes f the same rder fr each spectral line separated in wavenumber b R culd be recrded b varing d r The Reslving Pwer is then given b: R Inst Nw m/ d : m df Inst d Hence R. P. mf (7.4) Nte, F defines the effective number f interfering beams and m is the rder f interference. Alternativel, F determines the maximum effective path difference: R Maximum path difference = (dcs) F and (dcs m) S Maximum path difference mf i.e. the Reslving Pwer is the number f wavelengths in the maximum path difference.

43 7.7 Practical matters 7.7. Designing a Fabr-Pert (a) FSR: The FSR is small s F.P.s are used mstl t determine small wavelength differences. Suppse a surce emits spectral cmpnents f width ver a small range S. We will require FSR S. This determines the spacing d : d S r d (b) Finesse (Reflectivit f mirrrs). This determines the sharpness f the fringes i.e. the instrument width. We require Inst c Hence F d c The required reflectivit R is then fund frm r S F FSR c R F (7.5) ( R) 7.7. Centre spt scanning The pin-hle admitting the centre spt must be chsen t ptimize reslutin and light thrughput. T large and we lse reslutin; t small and we waste light and reduce signal-t-nise rati. We need t calculate the radius f the first fringe awa frm the central fringe: m cs m d If m th fringe is the central fringe, m 0 and s m d. The next fringe has angular radius: m cs d The fringe radius in fcal plane f lens f fcal length f: m fm This sets the maximum radius f the pinhle t be used. C Limitatins n Finesse The sharpness f the fringes is affected if the plates are nt perfectl flat. A bump f in height is visited effectivel 0 times if the reflectivit finesse is 0 and thus the path difference is altered b If the flatness is x it is nt wrthwhile making the reflectivit finesse > x /.

44 We assumed T ( R) i.e. n absrptin. In practice, hwever: R T A where A is the absrptin cefficient f the catings. The cefficient in equatin (8.) mdifies the transmitted intensit: T ( R) R A R A R Increasing R 00% means ( R) A and the cefficient in the Air functin: T I 0 0 ( R ) i.e. the intensit transmitted t the fringes tends t zer. Instrument functin and instrument width The instrument functin is the "mathematical" functin that describes the shape f the spectrum prduced in respnse t a delta-functin r mnchrmatic input. Fr example, in the case f a diffractin grating spectrmeter this is a functin made up frm a rati f sine functins: sin (N/)/sin (/). This functin describes the shape f the interference pattern i.e. a spectral line. The instrument width is the width f this functin defined in sme (arbitrar!) wa e.g. in the case f the diffractin grating we chse the separatin f the minima n either side f the peak f each rder. The pattern prduced b a diffractin grating spectrgraph cnsists f lines, smetimes knwn as spectral lines, that are simpl images f the entrance slit frmed b light f different wavelengths. In the case f the Michelsn interfermeter the instrument des nt prduce a "spectral line" directl. Rather the spectral line has t be calculated b finding the Furier Transfrm (F.T.) f the interfergram. Fr a mnchrmatic input wave the interfergram wuld be a sine wave ging n t infinit! The F.T. and hence the instrument functin wuld then be a delta functin. In practice hwever there is a limit t hw lng the interfergram can be - set b the maximum displacement f the mirrr. S the interfergram is a sine wave f finite length. The instrument functin wuld then be the F.T. f this finite sine wave, - brader than a delta functin and having sme finite width. This functin hwever is rarel used. Instead we usuall fcus n the effective spectral width that the instrument prduces fr a mnchrmatic wave - the instrument width - and this is fund frm the inverse f the maximum path difference - in units f reciprcal metres. S because different instruments prduce different shapes f "spectral lines" frm their interference patterns their instrument functins are different - s it's hard t cmpare them. Instead we usuall refer t the instrument width f each tpe f device - Grating, Michlesn r Fabr-Pert etc... as this allws a mre practical wa f cmparing their usefulness fr reslving spectral lines.

45 8. Reflectin at dielectric surfaces and bundaries 8. lectrmagnetic waves at dielectric bundaries Maxwell's equatins lead t a wave equatin fr electric and magnetic fields rr t H H rr t Slutins are f the frm: exp ik.r t Frm Maxwell's equatins we als have: H r t Frm which we find: r H where n is the refractive index f the medium and r r n H, H : is the impedance f free space. The cnstants r and r characterise the respnse f the medium t the incident electric field. In general the are represented b cmplex tensrs. This is because the affect the amplitude and phase f the light wave in the medium and the tensr nature reflects their dependence n the vectr nature f the fields and the smmetr f the medium in which the prpagate. Fr the time being we will cnsider nl linear istrpic hmgeneus (LIH) media and use scalar quantities fr the relative permeabilit r and permittivit r. In these cases the refractive index is then als a scalar quantit. 8.. Reflectin and transmissin at nrmal incidence When a wave is incident n a bundar between tw different media the change in electrical and magnetic respnse amunts t a change in impedance t the travelling wave. At such impedance bundaries sme f the wave energ is reflected and the rest transmitted. We cnsider first nrmal incidence i.e. the angle between the prpagatin directin k and the nrmal t the surface separating the tw media is 0. In this case there is smmetr with respect t the axis f prpagatin: the respnse will be the same n matter what the directin f the electric field (the magnetic field H is rthgnal t bth and k.) Thus the vectr nature f the fields ma be ignred in this case. When, hwever, the wave is incident bliquel at a bundar, the smmetr is brken and the directin f the and H fields must be cnsidered i.e. we must take accunt f the vectr nature f the fields and we will deal with this case later. At an bundar the fields must satisf bundar cnditins that are determined b cnservatin laws fr the electr-magnetic fields. The case f nrmal incidence at the interface f tw media f different refractive index n and n is illustrated in figure 8..

46 n n.n Figure 8. Reflectin f an electrmagnetic wave incident nrmall frm medium f refractive index n n a medium f index n Bundar cnditins demand that the perpendicular cmpnent f D is cntinuus and the tangential cmpnents f and H are cntinuus. Incident and reflected electric ' field amplitudes are and respectivel. Cnsidering the bundar cnditins fr bth and H allws us t calculate the rati f the incident and reflected amplitudes: n r n The intensit reflectin cefficient is therefre: n n Fr an air/glass interface ( n. 5 ), R ~ 4%. n n n n R (8.) 8. Reflectin prperties f dielectric laers. The reflectin at a dielectric bundar ma be thught f as arising frm an impedance mis-match. The larger the mis-match the larger the reflected prprtin f the incident energ. The pssibilit arises f using tw bundaries with intermediate mis-matches t generate tw reflected waves. If the mis-matches can be arranged t give apprximatel equal amplitude reflected waves it ma be pssible t arrange their relative phases t prduce destructive interference fr the backward travelling waves i.e. an anti-reflectin device. 8.. Reflectin prperties f a single dielectric laer. Cnsider a wave incident frm air, refractive index n, n a dielectric laer f index n depsited n a substrate f refractive index n T. See figure 8.., H are incident ' ' electric and magnetic wave amplitudes respectivel in the air, and H the reflected, amplitudes;, H and ' ', H are incident and reflected amplitudes in the dielectric laer and T is the amplitude transmitted t substrate. The wave vectrs are k i.

47 (a) (b) k.k T.k T k.k n.n.n..n T Figure 8. Reflected and transmitted waves fr a wave incident nrmall frm medium f index n n a dielectric laer f thickness, index n n a substrate f index n T. At bundar (a) using n H : H (8.) H H H (8.3) n ( ) n ( ) (8.4) At bundar (b), has acquired a phase shift wing t prpagating acrss the thickness f the laer: n ik ik e e T (8.5) ik ik ( e e ) nt T (8.6) liminating and ' frm (8.), (8.4), (8.5) and (8.6): n T cs k i k T n sin (8.7) n ( ) in sin k n T cs k T (8.8) Writing: A cs k C in sin k B i sin k n D cs k (8.9) We find:

48 and T r t An An An Bnn Bn n T T T C Dn C Dn n Bn n C Dn T T T (8.0) (8.) Nw cnsider the case when / 4, a quarter-wave laer; k / : Fr / a half-wave laer; k : nnt n R r (8.) n n n R T n nt r (8.3) Nte that fr a half-wave laer the refractive index f the laer des nt appear in the reflectivit and the result is the same as fr an uncated surface. (This effect is similar t that f a half-wave sectin f a transmissin line.) An anti-reflectin (AR) cating can be made i.e. ne that minimizes the reflectin b selecting a dielectric material such that n n T n 0 frm equatin (8.). This requires n n n. Fr an air/glass bundar this is nt pssible, the clsest we can d is t T have n as lw as pssible e.g. MgF has n =.38 giving R ~ %. Imprved AR catings are made using multiple laers. Catings ma als be made t enhance the reflectivit i.e high reflectance mirrrs. n n T Reflectance laer laer laer Uncated glass wavelength nm Figure 8.3 Anti-reflectin dielectric catings. A single / 4 laer can reduce reflectin frm 4% t ~%. Further reductin at specific wavelength regins is achieved b additinal laers at the expense f increased reflectivit elsewhere. This enhanced reflectin at the blue and red end f the visible is respnsible fr the purple-ish hue r blming n camera r spectacle lenses.

49 8.. Multiple dielectric laers: matrix methd. Write equatins (8.7) and (8.8) in terms f r, i.e. ' and t, i.e. T r in matrix frm: r ( A Bn ) t n ( r) ( C Dn ) t A B r t n n C D n T T T (8.4) The characteristic matrix is A B M (8.5) C D The characteristic matrix fr a laer f index n m is 0 i / nm M m (8.6) inm 0 A stack f N laers has a characteristic matrix: M M M M... (8.7) Stack 3 M N k k n.n.n H n L.n n H n L.n T Figure 8.4 Multiple quarter-wave stack 8..3 High reflectance mirrrs A stack f dielectric laers f alternate high and lw index n H, n L, respectivel has the matrix: M HL nl / n 0 H n H 0 / n L (8.8) N such pairs has a matrix, N M HL frm which we find the values f A, B, C and D.

50 Hence frm equatin (8.0) we find the reflectivit f the cmpsite stack: n N T nh n nl R Stack N (8.9) nt nh n nl 8..4 Interference Filters A Fabr-Pert etaln structure ma be cnstructed frm tw high reflectance stacks separated b a laer that is r an integer multiple f The half-wave laer(s) acts as a spacer t determine the Free Spectral Range, FSR. The FSR will therefre be ver large such that nl ne transmissin peak ma lie in the visible regin f the spectrum. This is an interference filter. Narrwer range filters ma be made b increasing the spacer distance and increasing the reflectance. xtra peaks ma be eliminated using a brad band high r lw pass filter..n H n L.n H n L.n H n L.n H.n H.n H n L n L.n T Figure 8.5 Interference filter cnstructed using multiple dielectric laers cnsisting f tw high-reflectance stacks separated b a laer which acts as a spacer in the Fabr-Pert tpe interference device. The spacer ma be made in integer multiples f / t alter the FSR. 8.3 Reflectin and transmissin at blique incidence We nw cnsider the mre general case f a light wave striking a dielectric bundar at an angle f incidence that is nt 90. We define the plane f incidence as that plane cntaining the wave prpagatin vectr k and the nrmal t the dielectric bundar. This i( tnr/ c) is illustrated in figure 8.6. The incident wave has the frm: 0e, where r is the distance alng the prpagatin axis OP. In general the electric field vectr can lie at an angle arund the wave vectr k. Hwever we can alwas reslve this vectr int tw cmpnents: ne ling in (i.e. parallel t) the the plane f incidence, P, and ne perpendicular t this plane, S. [A light wave with its -vectr parallel t the plane f incidence is knwn as p-plarized light. When the -vectr is perpendicular t this plane it is s-plarized, frm the German senkrecht meaning perpendicular.] The H-vectrs will be perpendicular t the respective -vectrs in each case.

51 Figure 8.6 Wave with electric field incident at blique angle f incidence frm dielectric with refractive index n t dielectric with index n. The electric field has a cmpnent P parallel t the plane f incidence and S perpendicular t the plane f incidence Reflectin and transmissin f p-plarized light We cnsider first the case f p-plarized light. This is represented in figure 8.7 in which the plane f incidence (the x-z plane) is the plane f the paper. The incident, reflected and P P P transmitted -fields all lie in this plane and are dented, and respectivel, ling respectivel at angles and t the nrmal r z-axis. The bundar cnditins require: Figure 8.7 P cs i( tn xsin / c) P i( tnx sin / c) P i( tnxsin / c) e cs e cs e (8.0) and the H-field cmpnents ut f the paper are: H P H P H P (8.)

52 Since these cnditins are beed at all times the expnential terms must be identical. Hence, i.e this is Snell s law fr reflectin and refractin. n sin n sin, (8.) Frm sectin 8. we can write H n and the bundar cnditins then becme, P P ( P P P n ( ) n Frm this we find, using Snell s law (8.), P )cs cs P P cs sin cs sin sin cs sin cs sin( )cs( ) P P sin cs sin cs tan( ) sin cs sin cs tan( ) (8.3) (8.4) 8.3. Reflectin and transmissin f s-plarized light When the light is plarized such that the -vectr is perpendicular t the plane f incidence the H-fields will lie in this plane as shwn in figure 8.8. Figure 8.8. The -vectrs are ut f the plane f the figure Fllwing the same prcedure, using the bundar cnditins n and H we find the fllwing relatinships between the incident, reflected and transmitted amplitudes,

53 S S S S cs sin cs sin cs sin sin cs sin( ) cs sin sin cs sin( ) cs sin sin cs sin( ) (8.5) (8.6) quatins (8.3), (8.4), (8.5) and (8.6), are the Fresnel quatins. [nn-examinable] The allw us t predict the reflectin and transmissin cefficients fr light f varius plarizatins incident n a dielectric surface i.e. a bundar between tw different dielectric media such as air and glass. Figure 8.9 Intensit reflectin cefficients fr s-plarized and p-plarized light as functin f incidence angle frm air t glass. The dashed line is the average and represents the behaviur fr unplarized light. The Fresnel equatins shw that the reflectin cefficients fr bth s- and p-plarized light var with angle f incidence as shwn in figure Deductins frm Fresnel s equatins 8.4. Brewsters Angle The amplitude reflectin cefficient, r, is given fr p-plarized light b equatin (8.4), P P tan( ) r tan( ) (8.4)

54 We can see that as ( ) appraches /, r will tend t zer [ tan( ) ]. In the limit( ), then cs sin. The angle f incidence at which this cnditin is satisfied is knwn as Brewster s angle B and, frm (8.), n n sin B sin cs B n n r tan n B (8.7) n Thus fr p-plarized light incident at angle B there will be n reflected wave. Anther cnsequence f this is that unplarized light, which cnsists, as we will see later, f waves with -vectrs varing randml in all pssible rientatins, will becme plane plarized with its -vectr perpendicular t the plane f incidence. This is because an - vectr ma be cmpsed f s- and p-plarizatins and nl the s-plarizatin will be reflected. An imprtant applicatin f Brewster s angle is in enabling p-plarized light t suffer n reflectin lsses when passing thrugh a glass windw. This is ver useful fr minimizing reflectin lsses at windw surfaces fr intense laser light Phase changes n reflectin We first cnsider the predictins f the Fresnel equatins fr nrmal incidence. Fr = 0, using (8.) we find, Hence, Similarl we find, P P P P P P cs sin sin cs sin cs cs ( n n)sin sin cs sin cs ( n n) sin cs sin cs n n n n n n n ( n n n n) cs cs (8.3a) (8.4a) S n (8.5a) S n n S S n n (8.6a) n n There appears t be a discrepanc between the equatins fr the reflected light plarized in rthgnal planes, (8.4a) and (8.6a) since, at nrmal incidence, the tw planes are indistinguishable, et the have ppsite signs! The sign discrepanc arises because we P P have taken the incident and reflected fields, and, t be in the ppsite directins

55 P (see figure 8.7). Assuming n > n and since the rati P / is psitive, ur ther predicts that there is a phase shift f in the reflected -vectr relative t the incident wave. Nte that as the incident and reflected H-vectrs are in the same directin there is n phase shift f the H-field. In the case f the s-plarized -vectrs in figure 8.8, the are bth perpendicular t the plane f the paper but in ppsite directins i.e. crrespnding t a -phase shift. In the case n < n there is n phase shift in but a -phase shift ccurs in the H- vectr. In bth cases there is n phase shift in the transmitted wave Ttal (internal) reflectin and evanescent waves When a wave is incident frm a medium f index n bliquel n a less dense medium, index n, as shwn in figure 8.0, we knw that if the angle f incidence exceeds a critical value crit, given b crit sin n n, then the wave is ttall reflected. In this case the angle in the less dense medium is an imaginar quantit, leading t phase shifts that lie between 0 and and are different fr and its crrespnding H. The analsis is tedius and ff-sllabus (thankfull!) but leads t the fllwing predictins.. The amplitude f the reflected beam and incident beams are equal ttal reflectin.. There is a transmitted beam with the fllwing characteristics: (a) its wave velcit is v /sin (b) it travels parallel t the interface (c) its amplitude decas expnentiall with distance perpendicular t the surface (d) it is neither a plane nr a transverse wave it has a cmpnent alng the surface (e) its Pnting vectr is zer. Figure 8.0 Internal reflectin, the evanescent wave is shwn as dtted lines When a wave is incident frm a medium f index n bliquel n a less dense medium, index n, as shwn in figure 8.0, we knw that if the angle f incidence exceeds a This, slightl msterius, transmitted wave is knwn as an evanescent wave. Its presence can