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

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1 OPTICS: the science f light nd year Physics FHS A P. wart Intrductin and structure f the curse. The study 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 study the universe frm the very distant t the very small. Newtn intrduced a scientific study als f the nature f light itself. Tday Optics remains a key element f mdern science, nt nly as an enabling technlgy, but in Quantum Optics, as a means f testing ur fundamental understanding f Quantum Thery and the nature f reality itself. Gemetrical ptics, studied in the first year, ignred the wave nature f light and s, in this curse, we fcus particularly n Physical Optics where the primary 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 elementary diffractin effects befre presenting, in chapter, the basics f scalar diffractin thery. By using scalar thery 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 thery allws us t treat mathematically the prpagatin f light and the effects f bstructins r restrictive apertures in its path. We then intrduce a very pwerful mathematical tl, the Furier transfrm and shw hw this can help in slving difficult diffractin prblems. Furier methds are used very widely in physics and recgnise the inter-relatin f variables in different dimensins such as time and frequency r space and spatial frequency. The latter cncept will be useful t us in understanding the frmatin f images in ptical systems. Having established the mathematical basis fr describing light we turn t methds f analysing the spectral cntent f light. The spectrum f light is the primary link between ptics and atmic physics and ther sciences such as astrphysics. The basis fr almst all instruments fr spectral analysis is, again, interference. The familiar Yung s slit, tw-beam, interference effect in which the interference arises frm divisin f the wave-frnt is generalised t multiple slits in the diffractin grating spectrmeter. The alternative methd f prducing interference, by 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 Fabry-Pert interfermeter. These devices are imprtant tls and play a key rle in mdern laser physics 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 physics f wave reflectin at bundaries can be engineered t prduce surfaces with high r partial reflectivity r even n reflectivity at all. Finally 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 linearly plarized light r rtating in space in the case f elliptically r circularly plarized light. We will study hw t prduce, manipulate and analyse the state f plarizatin f light.

2 . Waves and Diffractin. Mathematical descriptin f a wave A wave is a peridically repeating disturbance in the value f sme quantity. In the case f light this scillatin is f the electric r magnetic field strength r amplitude. Figure. u u cs( kzt r u u e e i i( kzt (. Where u is the amplitude, is the angular frequency and k is the wave number fr the wave scillatin f perid T and wavelength. The wave number k is a scalar but is smetimes used as a vectr, k, t indicate the directin f the wave prpagatin. t : phase change with time, kz : phase change with distance, k = : arbitrary initial phase 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 The additin f amplitudes frm tw surces gives interference e.g. Yung's slits: r r P d dsin D Figure. Yung's slits We cnsider tw slits separated by d illuminated by mnchrmatic plane waves Amplitude u p at a pint P a large distance, D, frm the slits

3 Imaginary u i( kr t u i( kr t u p e e r r (. The factrs f /r and /r accunt fr the fall-ff in amplitude accrding t the inverse square law fr the intensity f the wave. Putting (r - r = d sin, r ~ r = r, the intensity is:.3 Phasrs I p u 4 r cs ( kdsin (.3 The amplitude f a wave is represented by the length f a vectr n an Argand diagram. The phase f the wave is represented by the angle f the vectr relative t the Real axis f the Argand diagram. S that, as the phase changes with time r distance, the directin f the phasr will rtate in the plane f the real and imaginary axes. 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: Phase difference, wing t path difference d sin : kdsin Resultant amplitude is then u u p cs( / r The intensity is therefre: u I p r u 4 r cs ( kdsin (.4 (.5.4 Diffractin frm a finite slit A mnchrmatic plane wave incident n aperture f width a, is bserved in a plane at large distance D frm aperture. The amplitude in the plane f aperture is u per unit length.

4 y +a/ dy r + ysin r P -a/ ysin D Figure.5 Cntributins t amplitude at P frm elements dy in slit. An infinitesimal element f length dy at psitin y cntributes at P an amplitude: udy ( y e i r The phase factr ( y k( r ysin. The ttal amplitude at P arising frm all cntributins acrss the aperture: u p u e r a / ikr a / e ik sin. y dy (.6 The intensity is then: I p I(0 sinc where ka sin sinc ( Figure.6 Intensity pattern frm single slit, I p = I(0sinc. The first minimum is at, asin Hence angular width f diffractin peak is: a (.7 This result represents the basic physics f all wave diffractin phenmena.

5 .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 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 plygn. / 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 intensity at : I( = I(0 sinc (/ = I(0 sinc (.8

6 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:.6 Diffractin in dimensins a (.9 Recall that the amplitude resulting frm a plane wave illuminatin f an aperture f the frm f a slit f width a in the y - directin : u p u e r a / ikr a / e ik sin. y dy (.0 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 p u e r b/ ikr b/ e ik sin. x dx In - D we have : u p e ikr b / a / b / a / u( x, y. e ik (sin. xsin. y dxdy (. y x z Figure.0 General -D aperture in x,y plane. u(x,y 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: (. a.

7 Intensity y x Figure.. Pint Spread Functin fr circular aperture. A pint surce imaged by 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.

8 . Fraunhfer Diffractin S far we have cnsidered diffractin by (a Apertures r slits illuminated by plane waves (b Observatin at a large distance where the phase difference between cntributins frm secndary surces in the diffracting plane separated by y is given t a gd apprximatin by: k sin.y (. These are special cases where the phase difference is a linear functin f the psitin y 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. By linear we mean that the wave frnt deviates frm a plane wave by 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. ( R R a fr R 0a / Alternatively, Fraunhfer diffractin is the diffractin bserved in the image plane f an ptical system. Figure. Fraunhfer cnditin fr plane waves: image is at infinity as surce is at fcal length frm lens. O A B C P 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 will be bserved at P if BC

9 O P u v P O quivalent lens system 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 by an equivalent system where diffractin f plane waves ccurs. Nte hwever that this means plane waves are nt necessary t bserve Fraunhfer diffractin. The key criterin is that... the phase varies linearly 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. (a O P (b O P Figure.4 quivalent lens system 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. Huygens secndary surces in the wave frnt radiate t a pint P in the secnd plane.

10 ds n r P -z 0 z Plane wave surface Figure.5 Huygens secndary 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 bliquity factr - this accunts fr the fact that the wave prpagates nly in the frward directin. n is a unit vectr nrmal t the wave frnt and is a prprtinality cnstant - t be determined. We determine by a self-cnsistency 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 lying n a nrmal thrugh P. We nw cnstruct elements f the wavefrnt f equal area A centred n O. n r n q P Figure.6 Cnstructin f elements f equal area n plane wavefrnt. The first element is a circle, the n th is an annulus f uter radius ( n n A Cnsequently the difference in distance r frm successive elements t P is cnstant A r rn rn q (.3 n

11 Therefre the phase difference between waves frm successive elements is als cnstant: A q (.4 Hence we may 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 by these elements is the First Half Perid Zne, st HPZ. Figure.7 Cnstructin f the First Half Perid Zne The difference in path-length frm the uter element f the st HPZ t P and frm O t P is /. / q O Figure.8 Phase shift f arises at edge f st HPZ The radius f the st HPZ is is given by: q q P

12 Recalling ur diffractin integral we write the cntributin t the amplitude at P frm the st HPZ: u u q (.5 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 imaginary 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 exactly zer wing t the term /r (inverse square law fr intensity and the bliquity 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 Adding cntributins frm the whle wave (integrating ver infinite surface gives resultant equal t ½ the st HPZ. Therefre R iu Self-cnsistency demands that this wave at P matches the riginal wave at O: Hence u u iu i p i uds (n,r e r ikr (.6 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 by the bundary f the aperture. We simplify by: Ignring the bliquity factr i.e. put ( n, r, Restricting t ne dimensin: ds dx, Ignring the r term by cnsidering nly a small range f r, ikr ikr ik sin x Using the Fraunhfer cnditin: e e e, ikr Absrbing e int the cnstant f prprtinality: Since the integral will be zer wherever the amplitude functin u(x is zer the limits f integratin can be safely extended t infinity. The amplitude u p as a functin f angle is then, with k sin : u p We nte that A( is the Furier transfrm f u(x. Imprtant result: ix A( u( x e dx, (3. The Fraunhfer diffractin pattern is the Furier transfrm f the amplitude functin f the diffracting aperture. Mre precisely: 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: u( x A( e i x 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 by: 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( ( Sme useful Furier transfrms and cnvlutins (a We can represent a wave f cnstant frequency as a functin f time t. v( t V e it F. T.{ v( t} V( V ( (3.6 i.e. V( represents the spectrum f a mnchrmatic wave f frequency and is a delta functin in frequency space. Figure 3. A wave f cnstant frequency (mnchrmatic and its Furier transfrm Alternatively the inverse transfrm relatins allw us t represent the F.T. f a delta functin: it v( t V ( t t as inverse F. T.{ v( t} V( V e (3.7

15 (b The duble slit functin, i.e. tw delta-functins separated by d : (c A cmb f delta functins: d vb ( x ( x Vb ( cs( d (3.8 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 (3.9 where ( N x s The factr e i is simply the cnsequence f starting ur cmb at x = 0. This factr can be eliminated by shifting ur cmb t sit symmetrically 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 (3.0 [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 Analysis A peridic functin V(t may be represented by a Furier series. sin( cs( ( t p s t p c c t V p p p p (3. V(t is the result f synthesis f the set f Furier cmpnents. Furier analysis is the reverse prcess - finding the cmpnents (amplitude and phase that make up V(t. The cefficients are fund by 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: t e t V A e A t V t ip p t ip p p d ( ( 0 (3. This last expressin represents a Furier transfrm - suggesting that this peratin analyses the functin V(t t find the amplitudes f the Furier cmpnents A p.

17 3.5 Spatial frequencies Cnsider a plane wave falling nrmally n an infinite screen with amplitude transmissin functin: u( x sin( s x i.e. a grating with peridic pattern f width d s (3.3 This defines the spatial frequency: s d (3.4 The Fraunhfer diffractin pattern is then: where k sin. We find: A( u( x e ix dx A( 0 except fr 0, (3.5 i.e. sin 0 r d The sinusidal grating has a Fraunhfer diffractin pattern cnsisting f zer rder and + first rders d /. / s An additinal spatial frequency n will lead t additinal first rders at n /. [Nte: a finite screen will result in each rder being spread by the diffractin pattern f the finite aperture, i.e. the spread functin f the aperture.] 3.6 Abbé thery f imaging We cnsider an bject cnsisting f an infinite screen having a sinusidal transmissin described by 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 by 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 by 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. a v D s d v(x

18 Zer and first rder pints act as cherent surces giving tw-beam interference at psitins beynd the fcal plane. In the image plane, distance v frm the lens, the interference pattern is maximally sharp, v = f + D. The interference pattern is a sinusidal fringe system with spacing: D d a (3.6 Frm gemetry d d u v Hence: u v f (3.7 Fr a finite grating the pints will be spread by 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.] Any bject amplitude distributin may be synthesised by a set f sinusidal functins. ach Furier cmpnent with a specific spatial frequency 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 may 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 nly 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 partially fulfilled even fr white light illuminatin.] 3.7 The Cmpund Micrscpe Figure 3.4 shws the arrangement f the cmpund micrscpe. Basically a very 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 eye-piece is basically a simple magnifier used t view the real image which is lcated at the fcal length f the eyepiece giving a virtual image at infinity. This allws viewing with minimum eyestrain. 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 by the bjective lens. sin max d min (3.8

19 Spatial frequencies having dimensins smaller than dmin will diffract t larger angles, miss the bjective and thus nt appear in the image. The minimum spatial dimensin d min Figure 3.4 The Cmpund Micrscpe. The bject at distance u frm bjective with fcal length f is imaged at distance v. Real image is at fcal length f frm eyepiece giving angular magnificatin / where is the angle subtended by the real image if it was at the near pint f the eye, distance D. Apprximately, u = f O and v = L, the length f the tube. In this apprximatin the magnificatin is M DL f O f that can be reslved may be increased by immersing the bjective and bject in il f refractive index n ; the il immersin bjective: n sin max d n sin max is the Numerical Aperture and defines the ultimate reslutin f the device. min F max Figure 3.5. First rder diffracted waves frm spatial structures < d min are cllected by the lens and interfere with zer rder waves in the image plane 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 by the f/n. f the ptical system 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 by 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 spectrscpy use interference t prduce a wavelength-dependent pattern. Measurement f these patterns, r fringes, allws us t infer the spectral cntent f the light. The interfering beams are prduced either by divisin f wavefrnt r by divisin f amplitude. The diffractin grating divides the wavefrnt int multiple beams. The Michelsn divides the amplitude int tw beams and the Fabry-Pert interfermeter divides the amplitude int multiple beams. It is imprtant t knw where t lk fr the fringes. S first 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 beynd the screen cntaining the tw slits. The fringes are nn-lcalized and usually bserved under the Fraunhfer cnditin. (b N-slit diffractin, the diffractin grating. t f Figure 4. Diffractin grating fringes. Again we usually bserve the Fraunhfer cnditin. A mnchrmatic plane wave is diffracted i.e. suffers cnstructive interference at angle. Parallel light interferes at infinity r in the fcal plane f a lens. The fringes are lcalized at infinity r in the image plane f the instrument.

21 4. Divisin f amplitude The interference may invlve tw beams (Michelsn r multiple beams (Fabry-Pert. The situatins are mdelled by reflectin f light frm a surce at tw surfaces. The surce may be a pint r extended and the surfaces may 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. This system is equivalent t -pint surces r Yung's slit situatin. Therefre the fringes are nn-lcalized fringes f equal thickness.. P P Figure 4.3 A pint surce O prvides images P, P' in reflecting surfaces frming a wedge. O (b Parallel P P O This is similar t the wedge situatin with -pint surces. The fringes are nn-lcalized fringes f equal inclinatin Figure 4.4 A pint surce reflected in tw parallel surfaces again prvides tw images P, P' 4.. xtended surce (a Wedge R P P R Figure 4.5. xtended surce OS prvides tw images PR and P'R' by reflectin at wedged reflecting surfaces. O S 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 may 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.

22 (b Parallel images t surce t=x path difference xcs t=x circular fringe cnstant Figure 4.6 Upper figure shws tw images f extended surce by reflectin in parallel slab f thickness t. Lwer figure shws fringes f equal inclinatin frmed in fcal plane f a lens by 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 infinity. These fringes are mre cnveniently bserved in the fcal plane f a lens. e.g. the eye. Reflecting surfaces separated by t lead t tw images separated by t r x = t. Parallel light at an angle f inclinatin t the axis frm equivalent pints n the effective 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 by p p x Hence radii f fringes in fcal plane f lens with fcal length f : f rp rp x (4.3 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 by parallel surfaces will be imprtant fr the Michelsn and Fabry-Pert interfermeters.

23 5 The diffractin grating spectrgraph 5. Interference pattern frm a diffractin grating Cnsider a plane wave f wavelength incident nrmally n a reflecting r transmitting grating f N slits separated by d. The amplitude cntributed by each slit is u and the intensity f the interference pattern is fund by adding amplitudes and taking the squared mdulus f the resultant. ( N = I 4 N = 0 Figure 5. Intensity pattern and assciated phasr diagram fr -slit interference I( 4u cs ( where d sin Principal maxima at 0, n, f intensity 4u. One minimum beween principal maxima. ( N = 3 N = 3 9 I 4 0 Figure 5. Phasr diagrams fr 3-slit interference and intensity pattern Using phasrs t find resultant amplitude (a 0, n Principal maxima f intensity 9u. Tw minima between principal maxima. (b / 3 Minimum / zer intensity (c Subsidiary maxima f intensity u (d 4 / 3 Minimum / zer intensity

24 (3 N = 4 Principal maxima at 0, n f intensity 6u. 3 minima between principal maxima. In general we have principal maxima at 0, n, intensity 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 intensity pattern Amplitude f N phasrs: Hence intensity: i i i( N A u ue ue... ue (5. N sin ( I( I(0 (5. sin ( 5. ffect f finite slit width Grating f N slits f width a separated by d is a cnvlutin f a cmb f N functins f(x with a single slit (tp-hat functin g(x: N f ( x p ( x pd 0 ; g( x, a fr { x h( x f ( x g( x a }; g( x 0, fr { x a } Using the Cnvlutin Therem with, F( F. T.{ f ( x}, G( F. T.{ g( x} and H( F. T.{ h( x} H( F(. G(

25 Hence H where kdsin and kasin sin ( sin ( (5.3 N ( I( I(0. sin ( ( 5.3 Diffractin grating perfrmance 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 simplicity we cnsider nrmal incidence n the grating. Then principal maxima at: d sin n ( Angular dispersin The angular separatin d between spectral cmpnents differing in wavelength by d d n (5.5 d d cs Reslving pwer d d n min (a N Figure 5.4 (a Principal maxima fr wavelength and fr d such that the phase shift fr the tw differs by the change in between the maxima and first minima. (b The same situatin pltted as a functin f diffractin angle. The angular width t the first minimum min equals the angular separatin between 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. min N p (b min

26 and d sin Angular width t first minimum min is fund frm d d cs (5.6 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: The reslutin criterin is: Hence the Reslving Pwer is: Free Spectral Range d n d d cs n d cs min nn (5.8 (5.9 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 (5.0 ( 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 :

27 ( where and satisfy the grating equatin d(sin sin n Reflected light Diffracted light (a Reflected light Diffracted light (b Figure 5.5 (a Diffractin angle Reflectin angle fr rdinary grating. (b Blazed grating reflects light at same angle as diffracted rder (a Order (b 0 (c Order Figure 5.6 (a Grating intensity pattern and single slit diffractin pattern. (b ffect f single slit diffractin envelpe n grating diffractin intensity fr unblazed grating. (c Grating intensity pattern fr blaze set t reflect light int nd rder.

28 5.5 ffect f slit width n reslutin and illuminatin Cnsider the imaging frming system 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 system t image slit f widthx s t image x i. (b Images f slit are spectrally dispersed by diffractin at grating. Slit is imaged at angle frm diffractin grating leading t freshrtening by cs. In a diffractin grating spectrgraph the image is viewed at the diffractin angle and s is freshrtened by cs. f xi x s f (5. cs The minimum reslvable wavelength difference, R, has an angular width R : n R R (5.3 d cs Wavelengths having difference R are separated in the image plane f lens f by x R : where we used R nn Reslutin is achieved prvided: f x R (5.4 Nd cs x i x R

29 The limiting slit width x s is then: f f xs r x s (5.5 Nd W 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 by verlap f images at different wavelengths : reslutin nt imprved beynd diffractin limit but image brightness is reduced.

30 6 The Michelsn (Furier Transfrm Interfermeter A tw-beam interference device in which the interfering beams are prduced by 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 arms differing in length by t. M is image f M in M giving effectively a pair f parallel reflecting surfaces illuminated by an extended surce as in figure 4.6. CP is a cmpensating plate ensuring beams have equal thickness f glass in bth arms. 6. Michelsn Interfermeter Distance frm beam splitter t mirrrs differs by 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 at p, where p is an integer. xcs p (6. On axis the rder f interference is p = x/. Symmetry gives circular fringes abut axis. The fringes are f equal inclinatin and lcalized at infinity. They 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. The tw-beam interference pattern is: where, the wavenumber. r f rp (6.3 x p I( x I(0 cs I( x I(0 cs x (6.4

31 Input spectrum x 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 by. The crrespnding wavenumbers are and and they 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 intensity: s I0( I0( I0( is the intensity f each interfergram at x = 0. Then I( x I0( cs xcs x (6.5 (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 visibility f fringes cycles t zer and back t unity fr equal intesity cmpnents. T reslve the cmplete cycle requires a path difference x max

32 This lks like an interfergram f a light surce with mean wavenumber multiplied by an envelpe functin cs x. This envelpe functin ges first t a zer when a peak f interfergram fr first cincides with a zer in the interfergram fr. The visibility (r cntrast f the fringes cycles t zer and back t unity; the tell-tale sign f the presence f the tw wavelength cmpnents. The number f fringes in the range cvering the cycle is determined by 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 cycle 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 cycle in the csine envelpe functin: min x max min (6.6 xmax This minimum reslvable wavenumber difference is the instrument functin as it represents the width f the spectrum prduced by the instrument fr a mnchrmatic wave. Inst (6.7 xmax Hence the Reslving Pwer RP is: xmax RP. ( The Furier Transfrm spectrmeter Inst In Figure 6. we see that the interfergram lks like the Furier transfrm f the intensity 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 intensity at x = 0 and the Secnd term is a sum f individual interfergrams. Replacing cmpnents with discrete wavenumbers by a cntinuus spectral distributin: i Inst i

33 I ( x I0 S( cs x. d 0 where S ( is the pwer spectrum f the surce. Nw S ( 0 fr 0, s secnd term may be written: F ( x S( cs x. d (6.9 S F(x is the csine Furier Transfrm f F. T.{ S( } I( x I0 Or S( F. T. I( x I0 (6.0 Apart frm a cnstant f prprtinality the Furier transfrm f the interfergram yields the Intensity r Pwer Spectrum f the surce. See Figure 6.. The Michelsn interfermeter effectively cmpares a wavetrain with a delayed 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 uncertainty 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 why the limit n the uncertainty 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: Inst (6. Maximum path difference between interfering beams 6.4 The Wiener-Khinchine Therem Nte: this tpic is NOT n the syllabus but is included here as an interesting theretical digressin. The recrded intensity I(x is the prduct f tw fields, (t and its delayed replica (t + integrated ver many cycles. (The delay x/c. The interfergram as a functin f the delay may be written: ( ( t ( t dt (6. Taking the integral frm t be t we define the Autcrrelatin Functin f the field ( ( t ( t dt (6.3

34 The Autcrrelatin Therem states that if a functin (t has a Furier Transfrm F( then F. T.{ ( } F( F (. F( (6.4 Nte the similarity between the Autcrrelatin therem and the Cnvlutin Therem. The physical analgue f the Autcrrelatin therem is the Wiener-Kinchine Therem. The Furier Transfrm f the autcrrelatin f a signal is the spectral pwer density f the signal The Michelsn interfergram is just the autcrrelatin f the light wave (signal. Nte that and are related by a factr c where c is the speed f light. 6.5 Fringe visibility Fringe visibility 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 visibility f the fringes by V I max I min I max I min (6.5 The fringe visibility cmes and ges peridically as the tw patterns get int and ut f step. The example shwn cnsisted f tw surces f equal intensity. The visibility varies between and 0. If hwever the tw cmpnents had different intensity 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 delay t a minimum value f I - I. Denting the intensities by I and I. ( ( 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 (6.6 min / V / V max Measuring the rati f the minimum t maximum fringe visibility V min / V max allws the rati f the tw intensities t be determined.

35 6.5. Fringe visibility, cherence and crrelatin When the surce cntains a cntinuus distributin f wavelengths/wavenumbers the visibility 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 delay. At large path differences there will be a cntinuus distributin f interfergrams with a range f phase differences that average t zer and n steady state fringes are visible. The path difference x intrduced that brings the visibility t zer is a measure f the wavenumber difference L acrss the width f the spectrum f the surce. L (6.7 x L is the spectral linewidth f the surce. All light surces have a finite spectral linewidth and may be thught f as emitting wavetrains f a finite average length. When these wavetrains are split in the Michelsn, and recmbined after a delay, interference will ccur nly 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 perfectly mnchrmatic surce (if it existed! wuld give an infinitely lng wavetrain and the visibility wuld be unity fr all values f x. The tw parts f the divided wavetrain in this case remain perfectly crrelated after any delay is intrduced. If the wavetrain has randm jumps in phase separated in time n average by say c then when the tw parts are recmbined after a delay d < c nly part f the wavetrains will still be crrelated. The wavetrains frm the surce stay crrelated with a delayed replica nly 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 self-crrelatin alng the length f the electrmagnetic wave emitted by the surce. (see sectin 4.3 Thus the Michelsn prvides a measure f the lngitudinal cherence f the surce. [Nte. Light surces may als be characterised by their transverse cherence. This is a measure f the 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. every pint n a sphere centred n the surce will have the same phase. Similarly a plane wave is defined as a wave riginating effectively frm a pint surce at infinity. Such a surce will prvide Yung s Slit interference n matter hw large the separatin f the slits. (The fringes, f curse, will get very tiny fr large separatins. If the slits are illuminated by 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 may 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 thus indicates the extent f the spatial crrelatin in the phase f the tw mnchrmatic surces i.e. this measures the transverse cherence in the light frm the extended surce.]

36 7. The Fabry-Pert interfermeter This instrument uses multiple beam interference by divisin f amplitude. Figure 7. shws a beam frm a pint n an extended surce incident n tw reflecting surfaces separated by a distance d. Nte that this distance is the ptical distance i.e. the prduct f refractive index n and physical 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 by partial reflectin at each surface resulting in a set f parallel beams having a relative phase shift intrduced by 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 infinity the fringes are f equal inclinatin and lcalized at infinity. In practice a lens is used and the fringes bserved in the fcal plane where they appear as a pattern f cncentric circular rings. 7. The Fabry-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 by parallel surfaces with amplitude reflectin and transmissin cefficients r i, t i respectively. Amplitude reflectin and transmissin cefficients fr the surfaces are r, t and r, t, respectively.the phase difference between successive beams is: An incident wave e -it given by: d cs (7. is transmitted as a sum f waves with amplitude and phase 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

37 it t 0tt e i rr e and multiplying by the cmplex cnjugate t find the transmitted Intensity: It t t 0t t r r rr 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 Airy Functin. See figure 7. I( Figure 7. The Airy functin shwing fringes f rder m, m+ as functin f. 7. Observing Fabry-Pert fringes m (m+ The Airy functin describes the shape f the interference fringes. Figure 7. shws the intensity 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 Airy pattern may be bserved as a functin f d,, r A system fr viewing many whle fringes is shwn in figure 7.3.

38 An extended surce f mnchrmatic light is used with a lens t frm the fringes n a screen. Light frm any 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. Fabry-Pert interfermeter Lens Screen d xtended Surce Lens Figure 7.3. Schematic diagram f arrangement t view Fabry-Pert fringes. Parallel light frm the Fabry-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 m cs m leads t: 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 may 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 linearly 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 by the half-intensity pint f the Airy functin i.e. I t / I0 / when

39 sin / The value f at this half-intensity pint is sin / differs frm an integer multiple f by a small angle s we have: / The full width at half maximum FWHM is then 4 The sharpness f the fringes may be defined as the rati f the separatin f fringes t the halfwidth FWHM and is dented by the Finesse F r F (7.8 R F (7.9 ( R S the sharpness f the fringes is determined by the reflectivity f the mirrr surfaces. 3 [Nte that. i.e. a quick check f the slutin f the quadratic equatin fr R!] F ~ ( R 7.4 The Instrumental 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 by the instrument fr mnchrmatic light. Fr n-axis fringes (cs: Hence d Inst d d d d d Inst and : F Inst (7. df

40 7.5 Free Spectral Range, FSR Figure 7.4 shws tw successive rders fr light having different wavenumbers, and th (. Orders are separated by a change in f The ( m rder f th wavenumber may verlap the m rder f ( i.e. changing the wavenumber by 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 Fabry-Pert fringes fr wavenumber and ( bserved in centrespt scanning mde. The m th -rder fringe f and ( appear at a slightly 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 by changing the plate separatin d. (Because changing d will change. The phase can be varied by changing d, r. See equatin (7.5. In figure 7.3 the different rders fr a given wavelength are made visible by the range f values f. If the surce emits different wavelengths, fringes f the same rder will have different radii n the screen. 7.6 Reslving Pwer The instrumental width may nw be expressed as: FSR Inst (7.3 F df

41 Tw mnchrmatic spectral lines differing in wavenumber by R are just reslved if their fringes are separated by 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 by R culd be recrded by varying d r The Reslving Pwer is then given by: 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. Alternatively, F determines the maximum effective path difference: R S Maximum path difference (d cs F Maximum path difference and mf (d cs m (7.5 i.e. the Reslving Pwer is the number f wavelengths in the maximum path difference.

42 7.7 Practical matters 7.7. Designing a Fabry-Pert (a FSR: The FSR is small s F.P.s are used mstly 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 S C (b Finesse (Reflectivity f mirrrs. This determines the sharpness f the fringes i.e. the instrument width. We require FSR Inst c r c F Hence F d c The required reflectivity R is then fund frm (7.9 F R ( 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 away 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: f m m This sets the maximum radius f the pinhle t be used.

43 7.7.3 Limitatins n Finesse The sharpness f the fringes is affected if the plates are nt perfectly flat. A bump f in height is visited effectively 0 times if the reflectivity finesse is 0 and thus the path difference is altered by If the flatness is x it is nt wrthwhile making the reflectivity finesse > x /. 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 intensity: T ( R R A R A R Increasing R 00% means ( R A and the cefficient in the Airy functin: T ( R 0 i.e. the intensity transmitted t the fringes tends t zer. I 0

44 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 H, : t r r t H H r r Slutins are f the frm: t i k.r exp Frm Maxwell's equatins we als have: t H r Frm which we find: H n H r r r where n is the refractive index f the medium and is the impedance f free space. Figure 8. Reflectin f an electrmagnetic wave incident nrmally frm medium f refractive index n n a medium f index n Bundary cnditins at the interface f tw media f different refractive index n and n demand that the perpendicular cmpnent f D is cntinuus and the tangential cmpnents f and H are cntinuus. Incident and reflected field amplitudes are and ' respectively. n n n n r The intensity reflectin cefficient is therefre: n n n n R (8. Fr an air/glass interface R ~ 4%. n n.n