1. Geometrical Optics

Transcription

1 . Gemetrical Optics OPTICS P. Ewart. Fermat's Principle Light has been studied fr a lng time. Archimedes and ther ancient Greek thinkers made riginal cntributins but we mentin here Hern f Alexandria (c AD) as he was the first t articulate what has cme t be knwn as Fermat's Principle. Fermat, stated his principle as "Light travelling between tw pints fllws a path taking the least time." The mdern, and mre crrect versin, is as fllws: "Light prpagating between tw pints fllws a path, r paths, fr which the time taken is an extremum." The principle has a theretical basis in the quantum thery f light that avids the questin f hw the light "knws" what directin t g in s that it will fllw the maximal path! [Basically the wave functin fr the light cnsists f all pssible paths but all, except the ne crrespnding t the classical path, destructively interfere wing t variatins in the phase ver the different paths.] Fermat s principle is the basis f Gemetrical ptics which ignres the wave nature f light. The principle may be used t derive Snell's Laws f reflectin and refractin. Optical path length OAP = L, given by: L = + / / ( x + h ) + [( d x) h ] Fr a maximum r minimum dl dx = 0 frm which we find x = d / Hence the incidence angle θ = reflectin angle φ : Snell's law f reflectin. Using a similar prcedure we can derive Snell's law f refractin: n = sinθ n sin where θ and θ are the angles between the light ray and the nrmal t the surface between media f refractive index n and n respectively. θ

2 Gemetrical ptics uses the effective rule f thumb that light travels in straight lines in a hmgeneus medium f unifrm refractive index. Deviatins ccur at bundaries between media f different refractive index r if the index varies in space. The path f light indicated by a ray can be pltted using Fermat s Principle r its mre useful frm as Snell s laws. This allws us t lcate images f bjects frmed when light travels thrugh cmplicated lens systems r, in the case f mirages, thrugh a medium f spatially varying refractive index.. Lenses and Principal Planes P Figure. First Principal Plane Back Fcal Plane P Figure. Thin lens frmula: Frnt Fcal Plane Secnd Principal Plane u + v = f u,v and f are measured t centre f lens f ``zer'' thickness. Fr an bject at infinity u = parallel rays are fcussed in the image plane where v = f. This defines the fcal plane f a thin lens. Fr a thick r cmpund lens (cmpsed f several individual lenses) the principal planes lcate the psitin f an equivalent thin lens. (See Figures. and.) The effective fcal length is the distance frm the principal plane t the assciated fcal plane.

3 .3 Cmpund lens systems.3. Telepht lens Principal Plane Fcal Plane Figure.3 f T.3. Wide angle lens Principal Plane Fcal Plane Figure.4 f W.3.3 Telescpe (Astrnmical) f O f E β Figure.5 Angular magnificatin: M = = f f E

4 .3.4 Telescpe (Galilean) β f O f E Figure.6 Angular magnificatin:.3.5 Telescpe (Newtnian) M = = f f E f f E β Figure.7 Angular magnificatin: M = = f f E f is the fcal length f the bjective mirrr and (fr a spherical mirrr surface) equals half the radius f curvature..3.6 Cmpund Micrscpe f O f E β u v Figure.8 The bject at distance u frm bjective with fcal length f is imaged at distance v. Real image is at fcal length f E frm eyepiece giving angular magnificatin β/α where α is the angle subtended by the real image if it was at the near pint f the eye.

5 .4 Illuminatin f ptical systems The brightness f an image is determined by the f-number (f/n.): f/n. = fcal length Diameter f Aperture Aperture stps determine the amunt f light reaching the image f an image frming ptical system. Image f bjective in eyepiece Figure.8 Size f bjective is the effective aperture stp in a telescpe. Fr lenses f given fcal length, the size (diameter) f the lenses are chsen t match the aperture f the image recrding system e.g. pupil f eye. Aperture stps fr a telescpe (r bincular) system will be chsen as fllws: Aperture stp (eye piece) pupil f eye ( 0 mm) Aperture stp (bjective) such that the size f the image f the bjective frmed by the eyepiece pupil f eye Aperture stps fr a camera lens are set by an iris diaphragm ver a range f f/n frm. t0. A small f/n. is a large aperture and allws in mre light. This then allws a shrter shutter time t expse the film (r CCD) - useful fr mving bjects. The disadvantage is a reduced depth f field (range f distances in fcus in the image). (a) (b) Figure.9 Field stp prvides even illuminatin t image frmed by eyepiece. Off axis light in the telescpe in Figure.9(a) spills past the eyepiece - the light t the image at this angle is reduced in intensity, the image darkens twards the edge. In Figure.9(b) all light at a given angle thrugh the bjective reaches the image. There is equal brightness acrss the image at the expense f the field f view. Field stps: these ensure unifrm illuminatin acrss the image by eliminating light at large ff-axis angles. Light frm the edges f the field f view reaches the image with the same intensity as light frm the centre.

6 Physical Optics. Waves and Diffractin. Mathematical descriptin f a wave u T λ t, x Figure. u = u cs( t kz ) r u = u e i e i( t kz) t : phase change with time, = T kz : phase change with distance, k = : arbitrary initial phase. Interference Additin f amplitudes frm tw surces gives interference e.g. Yung's slits: r r P d θ dsinθ D Figure. Yung's slits Tw slits separated by d illuminated by mnchrmatic plane waves Amplitude u p at a pint P a large distance, D, frm the slits u p = u r e i( t kr ) + u r e i( t kr ) Putting (r r )=dsin, r r = r, intensity is: I p = 4 u r cs ( kdsin )

7 .3 Phasrs Amplitude f wave is represented by length f a ``vectr'' n an Argand diagram. Phase f wave represented by angle f vectr relative t Real axis f the Argand diagram. The phasr is then: ue i Imaginary u δ Real Figure.3 Phasr diagram Example: Yung's slits. u/r δ/ δ u/r Figure.4 Phasr diagram fr tw slit prblem. Amplitude frm each slit n screen: Phase difference, wing t path difference d sin : = kdsin Resultant amplitude is then The intensity is therefre: u r u p u p = u r cs( /) I p = 4 u r cs ( kdsin ).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. y +a/ dy θ r + ysinθ r P -a/ ysinθ D Figure.5 Cntributins t amplitude at P frm elements dy in slit.

8 An infinitesimal element f length d y at psitin y cntributes at P an amplitude: u dy r e i (y) The phase factr (y) =k(r ± ysin ). The ttal amplitude at P arising frm all cntributins acrss the aperture: u p = u a/ r e ikr e ik sin.y dy a/ The intensity is then: I p = I(0) sinc β where β = ka sinθ sinc (β) π π -5 π 0 π 5 π 0 3π β Figure.6 Intensity pattern frm single slit, I p = I(0) sinc. π The first minimum is at β = π, π = asinθ λ Hence angular width θ f diffractin peak is: = a.5 Diffractin frm a finite slit: phasr treatment +a/ θ r P -a/ asinθ D Figure.7 Cnstructin shwing elements at extreme edges f aperture cntributing first and last phasrs

9 On axis, θ = 0 the phasr elements sum t R Off axis, θ 0 successive phase shifts between adjacent phasrs bend the phasr sum t frm a sectin f a regular plygn. θ=0 R P θ=0 / R P Figure.8 (a) Phasr diagram fr finite slit shwing resultant R fr θ = 0 and θ 0 The phase difference between first and last phasrs fr θ 0 is = 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. δ/ R R P δ Figure.8 (b) Phasr diagram in the limit as phasr elements 0. The amplitude at θ relative t the amplitude at θ = 0 : length f chrd length f arc =. Rsin( /) R. = sinc( /) Then the intensity at θ : I(θ) = I(0) sinc (δ/) = I(0) sinc β Phasr arc t first minimum Phasr arc t secnd minimum Figure.9 Phasr diagram shwing mimina fr increasing phase shift δ 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 δ = π angular radius is: = a

10 .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 y - directin : u p = u a/ r e ikr e ik sin.y dy a/ 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 b/ r e ikr e ik sin.x dx b/ In -D we have : u p e ikr b/ b/ a/ u(x,y).e ik(sin.x+sin.y) dxdy a/ y x θ φ 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: θ =. Intensity λ a 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 f. This is the Pint Spread Functin r instument functin analgus t the impulse functin f an electrical circuit giving its respnse t a δ-functin impulse.

11 3. 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. 3. 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 ρ a a ρ R surce R R bserving pint diffracting aperture Figure 3. Wavefrnts incident n and exiting frm a plane aperture. (R + ) = R + a fr ρ λ/0 R 0a / Alternatively, Fraunhfer diffractin is the diffractin bserved in the plane f an image frmed by an ptical system. P O A B C Figure 3. Fraunhfer cnditin fr plane waves: image is at infinity 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 λ/0

12 O P u v P O Equivalent lens system Figure 3.3 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 3.4 Equivalent lens system shwing that Frauhfer diffractin is independent f psitin f aperture

13 3. 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. ds n r P -z 0 z Plane wave surface Figure 3.5 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. u p = u ds r ( n,r ) e ikr u is the amplitude frm element f area d S. ( 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 is a prprtinality cnstant - t be determined. We determine α 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 = mλ 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 cnstruct elements f the wavefrnt f equal area δa centred n O. ρ n r n q P Figure 3.6 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

14 ( n+ n )= A Cnsequently the difference in distance δr frm successive elements t P is cnstant r = r n+ r n A q Therefre the phase difference between waves frm successive elements is als cnstant: = A q 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 ρ n increases. We als ignre the small change in η(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. ρ π / (q+ λ/) R π q Figure 3.7 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 /. λ/ ρ π q O q Figure 3.8 Phase shift f λ/ arises at edge f st HPZ P

15 The radius f the st HPZ is is given by: = q Recalling ur diffractin integral abve we may write the cntributin t the amplitude at P frm the st HPZ: u q = u Frm the phasr diagram, the amplitude frm the st HPZ is the length f the phasr arc, u. The resultant R π 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 = i u 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 η(n,r). Adding further elements gives a slw spiral. First Full-Perid Zne Resultant f n FPZs Figure 3.9 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 = st HPZ. Therefre R = i u Self-cnsistency demands that this wave at P matches the riginal wave at O: u = iαu λ

16 = i Hence u p = i u ds r ( n,r ) e ikr This is the Fresnel-Kirchff diffractin integral.

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18 4. Furier methds in Optics 4. 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 u p = i u ds r (n, r )e ikr We simplify as fllws: η ( n, r) = Restrict t ne dimensin: ds dx Ignre r term by cnsidering nly a small range f r. Use the Fraunhfer cnditin: e ikr = e ikr ik sin x e Absrb e ikr int the cnstant f prprtinality: The amplitude u p as a functin f angle θ is then: where β = k sinθ. u p A( ) = u(x)e i x dx We nte that A(β) is the Furier transfrm f u(x). Imprtant result: 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 β = k sinθ where θ 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

19 4. 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 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. 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( ) H( ) =F( ).G( ) 4.3 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 i t β Figure 4. A wave f cnstant frequency (mnchrmatic) and its Furier transfrm F.T.{v(t)} = V( ) =V ( ) i.e. V(β) represents the spectrum f a mnchrmatic wave f frequency β ο and is a delta functin in frequency space. Alternatively 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 i t

20 (b) The duble slit functin, i.e. tw delta-functins separated by d : v b (x) = (x ± d ) V b ( ) =cs( d) (c) A cmb f delta functins: V(x) x S x V( β) β Figure 4. A cmb f -functins and their transfrm The F.T. f v c (x) is: where N v c (x) = m=0 (x mx s ) V c ( ) =e i sin( N x s) sin( x s) = (N ) x s The factr α 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. (d) The tp-hat functin: v d (x) =fr x a v d (x) =0fr x > a V d ( ) = a sinc( a) [What wuld be the result if the tp-hat was shifted t sit between x = 0 and x = a?]

21 Nw sme useful cnvlutins: (e) The duble slit: v s (x) =v b (x) v d (x) (f) The grating functin: v g (x) =v c (x) v d (x) (g) The triangle functin: v Δ (x) =v d (x) v d (x) This is a self-cnvlutin. The self-cnvlutin is knwn als as the autcrrelatin functin. 4.4 Furier Analysis A peridic functin V(t) may be represented by a Furier series. V(t) =c + p= c p cs(p t)+ s p sin(p t) p= 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 τ f the scillatin. In general: s p = 0 V(t)sin(p t) dt c p = 0 V(t)cs(p t) dt c = 0 V(t) dt V(t) = A p e ip t p= A p = 0 V(t) e ip t dt 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.

22 4.5 Spatial frequencies Cnsider a plane wave falling nrmally n an infinite screen with amplitude transmissin functin: u( x) = + sin( ω x) i.e. a grating with peridic pattern f width This defines the spatial frequency: s d = s The Fraunhfer diffractin pattern is then: where β = k sinθ. We find: s = d A( ) = u(x)e i x dx A( β ) = 0 except fr β = 0, ± ω λ i.e. sinθ θ = 0 r ± d The sinusidal grating has a Fraunhfer diffractin pattern cnsisting f zer rder and + first rdersθ = ± λ / d = ± λω s / π. An additinal spatial frequency ω n will lead t additinal first rders aθ = ± λωn / π. [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.] 4.6 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. Furier plane s d u(x) θ a d v(x) f u Figure 4.3 Object u(x) imaged by lens t v(x). v D Diffractin rders are waves with parallel wave vectrs at angles θ = 0 and +λ/d. A lens brings these parallel waves t a fcus as pints in the fcal plane separated by a = fλ/d.

23 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 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: Frm gemetry Hence: d = D a d u = d v u + v = f 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. Each Furier cmpnent with a specific spatial frequency cntributes + rders t the diffractin pattern at specific angles θ t the axis. The aperture α f the lens and bject distance u determine the maximum angle θ max that may be cllected. Diffractin rders at angles greater than θ 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.] 4.7 The Cmpund Micrscpe Figure 4.4 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 θ max that can be cllected by the bjective lens. sinθ λ max = d min

24 f O f E β u v Figure 4.4 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 E frm eyepiece giving angular magnificatin / where is the angle subtended by the real image if it was at the near pint f the eye. Spatial frequencies having dimensins smaller than d min will lead t diffractin t larger angles, miss the bjective and thus nt appear in the image. The minimum spatial dimensin d min 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 min n sinθ max is the Numerical Aperture and defines the ultimate reslutin f the device. θ max Figure 4.5. First rder diffracted waves frm spatial structures < d min are cllected by 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 > θ max, miss the bjective and d nt interfere with zer rder in the image. 4.7 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 ~ λ/d A light is lst frm the image by diffractin. This is diffractin limited imaging. Fr nn-diffractin limited imaging: Image brightness Fr diffractin limited imaging: Image brightness d A 4 d A

25 8 5 Optical instruments and fringe lcalizatin Optical instruments fr spectrscpy use interference t prduce a wavelength-dependent pattern. 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. Befre lking at specific instruments we cnsider the general questin f fringe lcalizatin. 5. Divisin f wavefrnt (a) Tw-slit interference, Yung's Slits nn-lcalised fringes Figure 5. 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 θ θ Figure 5. 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. 5. 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. f

26 5.. Pint surce (a) Wedge. P P O Figure 5.3 A pint surce O prvides images P, P' in reflecting surfaces frming a wedge. This system 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 θ Figure 5.4 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. θ 5.. Extended surce (a) Wedge R R P P O S Figure 5.5. Extended surce OS prvides tw images PR and P'R' by reflectin at wedged reflecting surfaces. Each 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.

27 (b) Parallel images t surce t=x path difference xcsα circular fringe t=x cnstant α Figure 5.6 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 α t the axis frm equivalent pints n the effective surces are brught tgether in the fcal plane. The path difference is xcsα and the phase difference δ: α = x cs Bright fringes (cnstructive interference) ccurs when the phase difference δ = pπ (p = integer) r x cs α = pλ 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 : r p r p+ = f 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 by parallel surfaces will be imprtant fr the Michelsn and Fabry-Pert interfermeters.

28 6 The diffractin grating spectrgraph 6. Interference pattern frm a diffractin grating Cnsider a plane wave f wavelength λ 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 = N = I 4 δ = 0 δ = π δ = π 0 π π δ Figure 6. Intensity pattern and assciated phasr diagram fr -slit interference where I( ) =4u cs ( ) = d sin Principal maxima atδ = 0, nπ, f intensity 4u. minimum beween principal maxima. () N = 3 N = 3 9 I 4 0 π π δ δ = 0 δ = π/3 δ = π δ = 4π/3 δ = π Figure 6. Phasr diagrams fr 3-slit interference and intensity pattern Using phasrs t find resultant amplitude (a) δ = 0, nπ Principal maxima f intensity 9u. minima between principal maxima. (b) δ = π / 3 Minimum / zer intensity (c) δ = π Subsidiary maxima f intensity u (d) δ = 4π / 3 Minimum / zer intensity (3) N = 4 Principal maxima at δ = 0, nπ f intensity 6u. 3 minima between principa maxima.

29 In general: Principal maxima at δ = 0, nπ, intensity N. (N ) minima at and width f principal maxima N. N nπ N I 4 0 π π δ δ = 0 δ = π/n δ = 4π/N δ = mπ/ N Figure 6.3 Phasr diagrams fr N-slit interference and intensity pattern Amplitude f N phasrs: Hence intensity: 6. Effect f finite slit width δ A = u + ue i + ue i ue i(n ) I( ) =I(0) sin ( N ) sin ( ) Grating f N slits f width a separated by d is a cnvlutin f a cmb f N δ functins with a single slit (tp-hat functin): N f(x) = (x pd) ; g(x) =, fr { a x a }; g(x) =0, fr { x > a } p=0 Using the Cnvlutin Therem with, h(x) =f(x) g(x) F( ) =F.T.{f(x)}, G( ) =F.T.{g(x)} and H( ) =F.T.{h(x)}

30 H( ) =F( ).G( ) Hence where δ = kd sinθ and γ = kasinθ ) ( ) H( ) = I( ) =I(0) sin ( N ) ) sin ( ). sin ( 6.3 Diffractin grating perfrmance 6.3. The diffractin grating equatin The equatin fr I( ) gives the psitins f principal maxima, δ = 0, nπ, 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 ccur fr dsin = n 6.3. Angular dispersin The angular separatin dθ between spectral cmpnents differing in wavelength by dθ: d d = n dcs Reslving pwer λ λ+ dλ λ λ+ dλ nπ Δδ = min π N δ Δθ = min Δθ λ (a) (b) Figure 6.4 (a) Principal maxima fr wavelength λ and fr λ + dλ such that the phase shift δ fr the tw differs by the change in δ between the maxima and first minima. (b) The same situatin pltted as a functin f diffractin angle θ. The angular width t the first minimum Δθ min equals the angular separatin Δθ λ between the tw wavelengths. Principal maxima fr wavelength λ ccur fr a phase difference f δ = nπ. The change in phase difference δ between the maximum and the first minimum is Δδ min. θ p θ

31 Δ min =± N and = d sin Angular width t first minimum Δθ min is fund frm S d d = dcs Δ min = d cs.δ min The angular separatin Δθ λ f principal maxima fr λ and λ + Δλ is fund frm: d d = n dcs Δ = n dcs Δ The reslutin criterin is: Δ = Δ min Hence the Reslving Pwer is: Δ = nn Free Spectral Range n th rder f λ and (n + ) th rder f (λ + Δλ FSR ) lie at same angle θ. { n λ = d sinθ = ( n + )( λ Δλ) } Hence verlap ccurs fr these wavelengths at this angle. The Free Spectral Range is thus: Δ FSR = (n + ) Nte: the Reslving Pwer n and the FSR /n.

32 6.4 Blazed (reflectin) gratings The Blaze angle ξ is set t reflect light int the same directin as the diffracted rder f chice fr a given wavelength. Fr incident angle φ and diffracted angle θ the blaze angle will be : ξ = ( φ + θ ) where φ and θ satisfy the grating equatin d(sin ± sin ) =n Reflected light Diffracted light (a) φ φ θ ξ φ θ Reflected light Diffracted light (b) Figure 6.5 (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 6.6 (a) Grating intensity pattern and single slit diffractin pattern. (b) Effect 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.

33 6.5 Effect 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 Δx s has a width: Δx i = f f Δx s f f Δx s (a) f grating Δx s θ f (b) Figure 6.7 (a) Image frming system t image slit f width Δx s t image Δx i. (b) Images f slit are spectrally dispersed by diffractin at grating. Slit is imaged at angle θ frm diffractin grating leading t freshrtening by csθ. In a diffractin grating spectrgraph the image is viewed at the diffractin angleθ and s is freshrtened by csθ. Δx i = f f cs Δx s The minimum reslvable wavelength difference, Δλ R, has an angular width Δθ R : Δ R = n d cs Δ R Wavelengths having difference Δλ R are separated in the image plane f lens f by Δx R : Δx R = f Ndcs λ where we used Δ λ R = nn Reslutin is achieved prvided: Δx i Δx R

34 The limiting slit width Δx s is then: Δx s f Nd r Δx s f W Nte: the ptimum slit width is such that the diffractin pattern f the slit just fills the grating aperture, W = Nd. Δx s > Δx R : reslutin reduced by verlap f images at different wavelengths Δ x < Δ : reslutin nt imprved beynd diffractin limit but image brightness is reduced. s x R

35 7 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 7. The Michlesn interfermeter. The beam splitter BS sends light t mirrrs M and ' M in tw arms differing in length by t. M is image f M in M resulting effectively in a pair f parallel reflecting surfaces illuminated by an extended surce as in figure 5.6. CP is a cmpensating plate t ensure beams traverse equal thickness f glass in bth arms. 7. Michelsn Interfermeter Distance frm beam splitter t mirrrs differs by t in the tw paths, and α is the angle f interfering beams t the axis. Resulting phase difference between beams: Cnstructive interference at δ = pπ, where p is an integer. = t cs = x cs (7.) x cs = p On axis the rder f interference is p = x/λ. 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). Tw-beam interference pattern: where ν =, the wavenumber. λ r p r p+ = f x I( x) = I(0) cs I( x) = I(0) δ [ + cs πν x] (7.) (7.3)

36 ν Input spectrum Detectr signal Interfergram Figure 7. Input spectrum f mnchrmatic surce and resulting interfergram btained frm scanning Michelsn interfermeter. 7. Reslving Pwer f the Michelsn Spectrmeter. Cnsider that we wish t reslve tw wavelengths λ and λ that differ by Δλ. The crrespnding wavenumbers are ν and ν and they prvide tw independent interfergrams s the resultant is the sum f the tw: I(x) = I 0( )[ + cs x]+ I 0( )[ + cs x] x Let the tw cmpnents have equal intensity: s I 0 ( ) = I 0 ( ) = I 0 ( ) is the intensity f each interfergram at x = 0. Then I(x) =I 0 ( ) + cs + x cs x (7.4) (a) I( ν ) x (b) I( ν ) x x max (c) I( ν) x Figure 7.3 (a) Interfergram f surce cmpnent ν (b) interfergram f surce cmpnent ν. (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

37 + 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 Δ =. 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 Δ min that can be reslved is fund frm the value f x max giving the cycle in the csine envelpe functin: Δ min x max = Δ min = x max (7.5) 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. Hence the Reslving Pwer RP is: Δ Inst = x max (7.6) RP. = = Δ Inst Δ Inst = x max (7.7) 7.3 The Furier Transfrm spectrmeter Cmparing Figure 7. with figure 4. 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) = I 0( )[ + cs x]+ I 0( )[ + cs x] In the case f multiple discrete wavelengths: I(x) = i = i I 0( i )[ + cs i x] I 0( i )+ i I 0( i )cs i x

38 First term n r.h.s. is ½ I where I is the ttal intensity at x = 0 Secnd term n r.h.s. is sum f individual interfergrams. Replacing cmpnents with discrete wavenumbers by a cntinuus spectral distributin: I(x) = I S( )cs i x.d where S( ) is the pwer spectrum f the surce. Nw S( ) =0 fr < 0, s secnd term may be written: F(x) is the csine Furier Transfrm f S (ν ) F(x) = S( )cs i x.d (7.8) F.T.{S( )} = I(x) I 0 Or S( ) F.T. I(x) I 0 (7.9) Apart frm a cnstant f prprtinality the Furier transfrm f the interfergram yields the Intensity r Pwer Spectrum f the surce. See Figure 7.. 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 ν are Furier pairs r cnjugate variables.[see equatin (7.9)] This explains why the limit n the uncertainty f wavenumber (r wavelength) measurement Δν Inst is just the inverse f x max. In essence this explains the general rule fr all interfermeters including diffractin grating instruments that: Δ Inst = Maximum path difference between interfering beams 7.4 The Wiener-Khinchine Therem (7.0) 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, E(t) and its delayed replica E(t + τ ) integrated ver many cycles. (The delay τ = x/c. ) The interfergram as a functin f the delay may be written:

39 ( ) = E(t)E(t + )dt (7.) Taking the integral frm t + we define the Autcrrelatin Functin f the field t be Γ( ) = E(t)E(t + )dt (7.) The Autcrrelatin Therem states that if a functin E(t) has a Furier Transfrm F(ω) then F.T.{Γ( )} = F( ) = F ( ).F( ) (7.3) 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 ω and ν are related by a factr πc where c is the speed f light. 7.5 Fringe visibility Fringe visibility and relative intensities Figure 7.3 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 (7.4) 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 simply 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 = which leads t: I I I I + V V min max = (7.5) 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.

40 7.5. 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 Δ L acrss the width f the spectrum f the surce. Δ L is the spectral linewidth f the surce. Δ L x (7.6) A surce having a finite spectral linewidth i.e. every light surce (!) 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 τ c then when the tw parts are recmbined after a delay τ d < τ 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 τ 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) In ther wrds 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 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. 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 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.]

41 8. The Fabry-Pert interfermeter This instrument uses multiple beam interference by divisin f amplitude. Figure 8. 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 δ intrduced by the extra path dcsθ between successive reflectins which depends n the angle θ f the beams relative t the axis. (See sectin 5.. (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. 8. The Fabry-Pert interference pattern This is dne in all the text bks (cnsult fr details). The basic idea is as fllws: d E tr 6 e -i3δ tr 4 tr tr 5 tr 3 tr θ E tr 4 e -iδ E tr e -iδ E t E t Figure 8. Multiple beam interference f beams reflected and transmitted by parallel surfaces with amplitude reflectin and transmissin cefficients r i, ti respectively. Amplitude reflectin and transmissin cefficients fr the surfaces are r, t and r, t, respectively. The phase difference between successive beams is: π δ = d csθ (8.) λ An incident wave E e -iωt is transmitted as a sum f waves with amplitude and phase given by: E t E 0 t t e i t E 0 t t r r e i t E 0 t t r r e i t...etc. Taking the sum f this Gemetric Prgressin in r r e iδ

42 E t E 0 t t e i t r r e i and multiplying by the cmplex cnjugate t find the transmitted Intensity: I t E t E t E 0 t t r r r r cs writing E 0 I 0, r r R and t t T, and cs sin / : T I t = I 0 (8.) 4R ( R) + sin δ / ( R) If there is n absrptin in the reflecting surfaces T R then defining 4R ( R) = Φ (8.3) I t = I 0 + Φsin δ / (8.4) This is knwn as the Airy Functin. See figure 8. I( δ) mπ (m+)π δ Figure 8. The Airy functin shwing fringes f rder m, m+ as functin f. 8. Observing Fabry-Pert fringes The Airy functin describes the shape f the interference fringes. Figure 8. shws the intensity as a functin f phase shift δ. The fringes ccur each time δ is a multiple f π. π δ = d csθ = mπ (8.5) λ

43 m is an integer, the rder f the fringe. The fringes f the Airy pattern may be bserved by a system t vary d, λ, r θ. A system fr viewing many whle fringes is shwn in figure 8.3. 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 Extended Surce Lens Figure 8.3. Schematic diagram f arrangement t view Fabry-Pert fringes. Parallel light frm the Fabry-Pert is fcussed n the screen. Frm equatin (8.5) the m th fringe is at an angle θ m mλ csθ m = (8.6) d The angular separatin f the m th and (m + ) th fringe is Δθ m is small s θ m θ m+ Δ m d cnstant Therefre the fringes get clser tgether twards the utside f the pattern. The radius f the θ m fringe is mλ ρ ( λ) = fθ m = f cs (8.7) 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 λ. Fringes are prduced f rder m linearly prprtinal t d r ν, (/λ). This als has the advantage that all the available light is put int the detected fringe n axis.

44 8.3 Finesse The separatin f the fringes is π in δ-space, and the width f each fringe is defined by the halfintensity pint f the Airy functin i.e. I t /I 0 / when sin / The value f δ at this half-intensity pint is δ / sin / δ / differs frm an integer multiple f by a small angle s we have: / The full width at half maximum FWHM is then Δδ Δ 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 π π Φ F = = (8.8) Δδ r R F = π (8.9) ( R) S the sharpness f the fringes is determined by the reflectivity f the mirrr surfaces. [Nte that F~ 3. R This gives a quick check t ensure the quadratic equatin fr R has been slved crrectly!] 8.4 The Instrumental width The width f a fringe frmed in mnchrmatic light is the instrumental width: π Δ δ Inst = (8.0) F Δ Inst is the instrument width in terms f the apparent spread in wavenumber prduced by the instrument fr mnchrmatic light. Fr n-axis fringes (csθ = ): d d d d

45 Hence d Δ Inst d Δ Inst F and: Δν Inst = (8.) df 8.5 Free Spectral Range, FSR Figure 8.4 shws tw successive rders fr light having different wavenumbers, and ( Δ ). Orders are separated by a change in f π The m th rder f wavenumber may verlap the m th rder f ( Δ. i.e. changing the wavenumber by Δ mves a fringe t the psitin f the next rder f the riginal wavenumber. d m and ( Δ d m Δ d This wavenumber span is called the Free Spectral Range, FSR: Δν FSR = (8.) d ν ν Δν I( d) m th (m+) Figure 8.4 Fabry-Pert fringes fr wavenumber ν and ν - Δν bserved in centre-spt scanning mde. The m th -rder fringe f ν and ν - Δν appear at a slightly different values f the interfermeter spacing d. When the wavenumber difference Δν increases s that the m th rder fringe f ν - Δν verlaps the (m+) th the Free Spectral Range, FSR th rder f ν the wavenumber difference equals In figure 8.4 the different rders fr each wavelength (wavenumber) are made visible by changing the plate separatin d. (Because changing d will change δ). The phase δ can be varied by changing d, λ r θ. See equatin (8.5). In figure 8.3 the different rders fr a given wavelength are made visible by the range f values f θ. If the surce emits different wavelengths, fringes f the same rder will appear with different radius n the screen. d

46 8.6 Reslving Pwer The instrumental width may nw be expressed as: Δν FSR Δν Inst = = (8.3) F df Tw mnchrmatic spectral lines differing in wavenumber by Δ R are just reslved if their fringes are separated by the instrumental width: Δ R Δ Inst ν ν Δν R I( d) Δν Inst. Figure 8.5 Reslutin criterin: light f tw wavenumbers, Δ R is reslved when the separatin f fringes fr and Δ R is equal t the instrument width Δ Inst As in Figure 8.4 the fringes f the same rder fr each spectral line separated in wavenumber by Δ R culd be recrded by varying d r θ. The Reslving Pwer is then given by: Nw m/d : Δ R Δ Inst d Δ Inst m d df Hence ν R. P. = = mf (8.4) Δ ν R Nte, F defines the effective number f interfering beams and m is the rder f interference. Alternatively, F determines the maximum effective path difference: S Maximum path difference dcs F and dcs m Maximum path difference mf i.e. the Reslving Pwer is the number f wavelengths in the maximum path difference.

47 8.7 Practical matters 8.7. 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 Δ c ver a small range Δ S. We will require Δ FSR Δ S. This determines the spacing d : d < Δν S d Δ S r (b) Finesse (Reflectivity f mirrrs). This determines the sharpness f the fringes i.e. the instrument width. We require Δ Inst Δ c r Δ FSR F Δ c Hence F dδ c The required reflectivity R is then fund frm R F = π (8.5) ( R) 8.7. 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-tnise rati. We need t calculate the radius f the first fringe away frm the central fringe: cs m m d If m th fringe is the central fringe, θ m = 0 and s m= d/ λ The next fringe has angular radius: m cs d The fringe radius in fcal plane f lens f fcal length f: ρ fθ = m m Limitatins n Finesse The sharpness f the fringes is affected if the plates are nt perfectly flat. A bump f λ/0 in height is visited effectively 0 times if the reflectivity finesse is 0 and thus the path difference is altered by λ. If the flatness is λ/x it is therefre nt wrthwhile t make the reflectivity finesse > x/. We assumed T R i.e. n absrptin. In practice,