Chapter 16. Fraunhofer Diffraction

Transcription

1 Chapte 6. Faunhofe Diffaction

2 Faunhofe Appoimation Faunhofe Appoimation ( ) ( ) ( ) ( ) ( ) λ d d jk U j U ep,, Hugens-Fesnel Pinciple Faunhofe Appoimation : ( ) ( ) ( ) λ π λ d d j U j e e U k j jk ep,, ) ( ( ) ma k ) ( ) ( ) ( ) ( ) ( FT

3 Faunhofe diffaction Specific sot of diffaction fa-field diffaction plane wavefont Simple maths Faunhofe Diffaction

4 ( 참고 ) Fesnel Appoimation ( ) ( ) ( ) ( ) [ ] λ d d k j U j e U jk ep,, ( ) ( ) ( ) ( ) ( ) λ λ π d d e e U e j e U j k j k j jk,, ( ) ( ) ( ) f f k j k j jk Y X e U e j e U λ λ λ /, /, ), ( F ( ) ( )

5 Fesnel Diffaction This is most geneal fom of diffaction No estictions on optical laout nea-field diffaction cuved wavefont Analsis difficult Obstuction Sceen Fesnel Diffaction

6 6-. Faunhofe Diffaction fom a Single Slit Conside the geomet shown below. Assume that the slit is ve long in the diection pependicula to the page so that we can neglect diffaction effects in the pependicula diection.

7 Faunhofe Diffaction fom a Single Slit The contibution to the electic field amplitude at point P due to the wavelet emanating fom the element ds in the slit is given b de ( ω ) dep ep i k t et fo the souce element ds at s. Then fo an element de P de ( Δ) ep { i k( Δ) ωt } We can neglect the path diffeence Δ in the amplitude tem, but not in the phase tem. We let de E ds, whee E is the electic field amplitude, assumed unifom ove the width of the slit. The path diffeence Δ s sinθ. Substituting we obtain E b / ds dep ep{ i k ( s sinθ) ωt } EP ep i ( k ωt) ep( i k ssinθ) ds b/ ( ikssinθ) ep Integating we obtain EP ep i ( k ωt) b / iksinθ b /

8 Faunhofe Diffaction fom a Single Slit Evaluating with the integal limits we obtain ( iβ) ep( iβ) ep EP ep i( k ωt) iksinθ whee β kbsinθ Reaanging we obtain b EP ep i( k ωt) ep( iβ) ep( iβ) iβ ep i k ( ωt) b iβ E ( isinβ) ep i( k ωt) bsin β β The iadiance at point P is given b * b β β P P sinc β β I I sin c β I sin c ( kbsinθ ) sin sin I ε ce E ε c I I β

9 Faunhofe Diffaction fom a Single Slit The iadiance at point P is given b * b β β P P sinc β β sin sin I ε ce E ε c I I β sin β The sinc function is fo β, lim sinc β lim β β β The eoes of iadiance occu when sin β, o when β k bsinθ mπ m ±, ±, K ( ) I I sin c β I sin c kbsinθ

10 Faunhofe Diffaction fom a Single Slit In tems of the length on the obsevation sceen, f sin θ, and in tems of wavelength λ π / k, we can wite I I sin c β π π b β b λ f λ f β kbsinθ Zeoes in the iadiance patten will occu when π b mλ f mπ λ f b 3.47π The maimum in the iadiance patten is at β. Seconda maima ae found fom.46π d sin β cos β sin β β cos β sin β dβ β β β β.43π sin β β cos β tan β

11 Faunhofe Diffaction fom a Single Slit I I sin c β Note: - and -aes switched in book, Figs. 6-5a (hee) and Fig. 6- do not match.

12 The angula width of the cental maimum is defined as the angula sepaation Δθ between the fist minima on eithe side of the cental maimum, 6-. Beam Speading f sinθ θ W The fist min ima in the iadiance patten will occu when ( ± ) mλ f λ f λ Δθ b b b The width W of the diffaction patten thus inceases lineal with distance fom the slit, in the egions fa fom the slit whee Faunhofe diffaction applies WΔθ λ b

13 6-3. Rectangula Apetues When the length a and width b of the ectangula apetue ae compaable, a diffaction patten is obseved in both the - and - dimensions, govened in each dimension b the fomula we have alead developed. The iadiance patten is I ( α)( β) I sinc sinc whee, α k asinθ Zeoes in the iadiance patten ae obseved when mλ f mλ f o b a

14 Cicula Apetues E p E A Aea e isk sinθ da da ds R s R s E p E R A isk sin θ e R R v s / R, γ krsinθ s ds E p { } i v e v dv EAR E π AR J γ ( γ γ ) (the fist ode Bessel function of the fist kind)

15 Bessel Functions γ kr sin θ kdsinθ 3.83 (fist eo)

16 E p Faunhofe Diffaction fom Cicula Apetues: The Ai Patten E π γ ( AR J ) γ I / I() J ( γ ) when γ (o, at θ γ ( kd θ ) ( kdsinθ ) J sin I( θ ) I( ) ) Fist minimum in the Ai patten is at π D kdsinθ kdθ 3.83 θ λ θ Δθ. min λ D min γ kdsinθ

17 Ai Disc Ai pattens

18 Slit and Cicula Apetues Intensit Single slit (sinc ftn) Cicula apetue 3λ/D λ/d λ/d λ/d λ/d 3λ/D Sin θ

19 6-4. Resolution Abilit to discen fine details of object od Raleigh in 896» esolution is function of the Ai disc. Raleigh: imit of esolution» Two light souces must be sepaated b at least the diamete of fist dak band.» Called Raleigh Citeion

20 Raleigh Citeion Raleigh imit

21 Raleigh imit Resolution limit of a lens: min λf. D λ. NA.6λ NA (f focal length)

22 Faunhofe Diffaction fom a Double Slit Now fo the double slit we can imagine that we place an obstuction in the middle of the single slit. Then all that we have to do to calculate the field fom the double slit is to change the limits of integation. ( a b) / EP ep i( k ωt) ep( ikssinθ) ds ( ab) / ( ab) / ep i( k ωt) ep( ikssinθ) ds ( a b) / Integating we obtain ( a b) / ep( sin ) ep iks θ ( ikssinθ ) EP ep i( k ωt) iksinθ iksinθ ( ab) / ( ) ( ) ( ) ep i k ωt ik a b sinθ ik ab sinθ ep ep iksinθ ( ) sinθ ( ) ik ab ik a b sinθ ep ep ( ab) ( a b) / / E P ( ω ) bep i k t iβ { ep( iα ) ep( iβ) ep( iβ) ep( iα) ep( iβ) ep( iβ) } whee α k a sinθ and β k bsinθ

23 6-5. Faunhofe Diffaction fom a Double Slit But we know that ep α ep α cosα ( i ) ( i ) ( ) ( ) ep iβ ep iβ isin β Substituting we obtain E P ( ω ) bep i k t iβ ( cosα)( isinβ) The iadiance at point P is given b * b β P P ( ) β 4sin I ε ce E ε c 4cos α 4I cos 4 sin β α β whee I b ε c

24 Faunhofe Diffaction fom a Double Slit The iadiance at point P fom a double slit is given b the poduct of the diffaction patten fom single slit and the intefeence patten fom a double slit. I sin β 4I cos α β

25 Faunhofe Diffaction fom a Double Slit Single Slit Double Slit

26 6-6. Faunhofe Diffaction fom Man Slits (Gating) Now fo the multiple slits we just need to again change the limits of integation. Fo N even slits with width b evenl spaced a distance a apat, we can place the oigin of the coodinate sstem at the cente obstuction and label the slits with the inde j (Note that the diagam does not eactl coespond with this). { } j N/ ( j ) a b / EP ep i( k ωt) ep( ikssinθ) ds ( j ) ab / j P ( j ) a b / ( j ) ab / ep Integating we obtain ( ikssinθ) ( ikssinθ) ( j ) ( j ) ( ikssinθ) ( j ) a b / j N/ E ep ep ( ω ) j iksinθ ( j ) ab / E i k t ep iksinθ a b / ab / ds j N/ ( ) ( ) ( ) ep i k ωt ik j a b sin ik j a b sin θ θ ep ep iksinθ j ( ) θ ( ) ik j ab sin ik j ab sinθ ep ep

27 Substuting using α k asinθ and β k bsinθ and eaanging we obtain ( ω ) bep i k j N/ t EP i j i i i j i i iβ j We can ewite this as E P ( ω ) bep i k t i β { ep ( ) α ep( β) ep( β) ep( ( ) α) ep( β) ep( β) } j N/ { } ( sin i ) epi( j ) epi( j) j N/ sin β bep i( k ωt) Re ep i( j ) β j j { α } j N/ sin β bep i( k ωt) Re ep( iα) ep( i3α) ep( i5α) ep i( N ) α β j The last tem is a geometic seies that conveges to β α α { } j N/ j { ( iα) ( i α) ( i α) i( N ) α } Re ep ep 3 ep 5 ep sin Nα sinα The details of the last step ae outlined in the book. The iadiance at point P is given b * b sinβ IP cε EP EP cε sin Nα sin sin N I β α β α β sinα

28 Faunhofe Diffaction fom Multiple Slits The iadiance at point P is given b I sin β sin Nα I β sinα * E b sin β sin Nα sin β sin Nα P ε P P ε β sinα β sinα I c E E c I sin β β sin Nα sinα sin Nα When α mπ, the tem is a maimum. Fo this condition, fom ' Hospital ' s ule sinα N lim ( sin Nα ) d sin Nα cos lim d N Nα α lim ± N sinα d ( sinα ) cosα dα α mπ α mπ α mπ The pincipal maima the iadiance patten occu fo π pπ p α ka sinθ a sinθ mπ m, ±, ±, K λ N N m Fo lage N, the pincipal maima ae bight and well sepaated. This analsis gives us the gating equation, m θ mλ a sin θ mλ m

29 Diffaction gating equation a θ mλ m sin m m m m

30 Faunhofe Diffaction fom Multiple Slits N N 3 N 4 N 5