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Chapter 11. Fraunhofer Diffraction

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Chapter 11. Fraunhofer Diffraction ChapterChapter 11.11. FraunhoferFraunhofer Diffraction Diffraction Last lecture • Numerical aperture of optical fiber • Allowed modes in fibers • Attenuation • Modal distortion, Material dispersion, Waveguide dispersion This lecture • Diffraction from a single slit • Diffraction from apertures : rectangular, circular • Resolution : diffraction limit • Diffraction from multiple-slits DiffractionDiffraction regimesregimes Fraunhofer diffraction • Specific sort of diffraction – far-field diffraction – plane wavefront – Simpler maths FraunhoferFraunhofer DiffractionDiffraction Fresnel Diffraction • This is most general form of diffraction – No restrictions on optical layout • near-field diffraction • curved wavefront – Analysis difficult Obstruction Screen FresnelFresnel DiffractionDiffraction 11-1.11-1. FraunhoferFraunhofer Diffraction Diffraction fromfrom aa SingleSingle SlitSlit • Consider the geometry shown below. Assume that the slit is very long in the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction. Δ r0 The contribution to the electric field amplitude Δ at point P due to the wavelet emanating from the element ds in the slit is given by dE ⎛⎞0 r dEP =−⎜⎟exp ⎣⎦⎡⎤ i() krω t 0 ⎝⎠r Let r== r0 for the source element ds at s 0. Then for any element ⎛ dE ⎞ dE = 0 exp ikr⎡⎤+Δ −ω t P ⎜⎟⎜ ⎟ {}⎣⎦()0 ⎝ ()r0 +Δ ⎠ why? We can neglect the path differenceΔ in the amplitude term,. but not in the phase term We letdE0 = EL d s,, where EL is the electric field amplitude assumed uniform over the width of the slit . The path differenceΔ= s sinθ. Substituting we obtain ⎛⎞Eds ⎛⎞ E b /2 dE=+−=−L exp i⎡⎤ k() r s sinθω t EL exp ⎡⎤ i() kr ω t exp ( i k s sin θ ) ds P ⎜⎟{}⎣⎦00P ⎜⎟ ⎣⎦∫−b/2 ⎝⎠rr0 ⎝⎠0 b /2 ⎛⎞EL ⎡⎤exp()iks sinθ Integrating we obtain EP =−⎜⎟exp ⎡⎤ i() kr0 ω t ⎢⎥ rik⎣⎦sinθ ⎝⎠0 ⎣⎦−b /2 Evaluating with the integral limits we obtain ⎛⎞EL ⎡⎤exp()iiββ−− exp ( ) EikrtP =−⎜⎟exp ⎣⎦⎡⎤()0 ω ⎢⎥ ⎝⎠rik0 ⎣⎦sinθ where 1 βθ≡ kbsin 2 Rearranging we obtain ⎛⎞EL b EikrtiiP =−⎜⎟exp⎣⎦⎣⎡⎤⎡()0 ωββ exp() −− exp ( ) ⎦⎤ ⎝⎠ri0 2 β ⎛⎞EbL ⎛⎞EbL sin β =−⎜⎟exp ⎣⎦⎡⎤ikr()0 ω t ()2siniikrtβω=−⎜⎟ exp⎣⎦⎡⎤()0 ⎝⎠ri0 2 β ⎝⎠r0 β The irradiance at point P is given by 2 11⎛⎞Eb sinsin22ββ 2 1 I=εε cE E* == cL I 00PP ⎜⎟ 22 0 II= 0 sin c (β ), βθ=2 kb sin 22⎝⎠r0 ββ 2 1 II= 0 sin c (β ), βθ=2 kb sin sin β Thesinc function is 1 for ββ== 0, lim sinc lim= 1 ββ→→00β 1 The zeroes of irradiance occur whensinβ = 0, or whenβθπ= k bmsin = , m =± 1,± 2, 2 K 2λ The angular width of the central maximum : Δ≈θθ()sin − sin θ = mm=+11 =− b In terms of the length y on the observation screen, y ≅ f sinθ,2/,and in terms of wavelengthλπ= k we can write 12ππyby β ==b y 2 λλff Zeroes in the irradiance pattern will occur when f π by mfλ =⇒mπ y = 3.47π λ f b 2.46π The maximum in the irradiance pattern is at β = 0. Secondary maxima are found from 1.43π d ⎛⎞sinββββββ cos sin cos− sin ⎜⎟=−=22 =0 dββ⎝⎠ β β β 0 sin β ⇒=ββ =tan cos β Fraunhofer Diffraction pattern from a Single Slit 2 1 II= 0 sin c (2 kb sinθ ) 11-2.11-2. BeamBeam spreadingspreading duedue toto diffractiondiffraction ⎛⎞L WL=Δ=θλ2 ⎜⎟ ⎝⎠b 16-3.16-3. RectangularRectangular AperturesApertures When the length a and width b of the rectangular aperture are comparable, a diffraction pattern is observed inboth the x - and y - dimensions, governed in each dimension by the formula we have already developed : II= sinc2α sinc2β 0 ()() x where y 1 αθ= kasin 2 Zeroes in the irradiance pattern are observed when mfλλ mf yorx== ba SquareSquare AperturesApertures CircularCircular AperturesApertures E dA = xds E = A eisk sinθ dA p ∫∫ 2 r0 Area ⎛ x ⎞ R2 = s2 + ⎜ ⎟ ⎝ 2 ⎠ x = 2 R2 − s2 R 2EA isk sinθ 2 2 E p = e R − s ds ∫−R r0 v = s / R, γ = kRsinθ 2 2 1 2EAR iγv 2 2EAR ⎧πJ1(γ )⎫ E p = e 1− v dv = {}∫−1 ⎨ ⎬ r0 r0 ⎩ γ ⎭ (the first order Bessel function of the first kind) FraunhoferFraunhofer Diffraction Diffraction fromfrom CircularCircular Apertures:Apertures: BesselBessel FunctionsFunctions 1 γ = kRsinθ = kDsinθ = 3.832 (first zero) 2 FraunhoferFraunhofer Diffraction Diffraction fromfrom CircularCircular Apertures:Apertures: TheThe AiryAiry PatternPattern 2 2EAR ⎧πJ1(γ )⎫ E p = ⎨ ⎬ I()γ r0 ⎩ γ ⎭ ⎧ J (γ ) 1⎫ ⎨ 1 → ⎬ when γ → 0 (or, at θ = 0) ⎩ γ 2⎭ 2 ⎡⎤2J1 ()γ II()θ = () 0⎢⎥: Airy pattern ⎣⎦γ First minimum in the Airy pattern is at 11 ⎛⎞⎛⎞2π D 22kDsinθ ≅== kDθθ 3.83 ⎜⎟⎜⎟min ⎝⎠⎝⎠λ 2 λ θmin = Δθ 1 =1.22 2 D : Far-field angular radius 2 ⎡⎤2J1 ()γ II()θ = () 0⎢⎥ Airy pattern and Airy disc ⎣⎦γ Airy pattern and Airy disc Airy disc ComparisonComparison :: SlitSlit andand CircularCircular AperturesApertures Intensity Single slit (sinc function) Circular aperture (Airy function) 0 λ/D 2λ/D 3λ/D −3λ/D −2λ/D −λ/D Sin θ 16-4.16-4. ResolutionResolution • Ability to discern fine details of object – Lord Rayleigh in 1896 » resolution is a function of the Airy disc. » Two light sources must be separated by at least the diameter of first dark band. » Called Rayleigh Criterion Image blurring due to diffraction Rayleigh Criterion : Two light sources must be separated by at least the diameter of first dark band. separated Rayleigh limit confused RayleighRayleigh LimitLimit Resolution limit of a lens: ⎛⎞λ f xfmin=Δθ min =1.22⎜⎟ ⎝⎠D ⎛⎞λ 0.61λ ≈=1.22⎜⎟ ⎝⎠2NA NA (f = focal length) xmin ≈ λ The resolution of a microscope is roughly equal to the wavelength. 11-5.11-5. FraunhoferFraunhofer Diffraction Diffraction fromfrom DoubleDouble SlitsSlits Now for the double slit we can imagine that we place an obstruction in the middle of the single slit. Then all that we have to do to calculate the field from the double slit is to change the limits of integration. ()ab+ /2 ⎛⎞EL EikrtiksdsP =−exp⎡⎤()0 ωθ exp ( sin ) + ⎜⎟ ⎣⎦∫()ab− /2 ⎝⎠r0 −−()ab/2 ⎛⎞EL exp⎡⎤ikr()0 − ωθ t exp ( iks sin ) ds ⎜⎟ ⎣⎦∫−+()ab/2 ⎝⎠r0 Integrating we obtain ⎧⎫()ab+ /2 −−()ab/2 ⎛⎞EL ⎪⎪⎡⎤exp()iks sinθ ⎡⎤ exp() ikssinθ EikrtP =−⎜⎟exp ⎡⎤()0 ω ⎨⎬ + rik⎣⎦⎢⎥sinθ ⎢⎥iksinθ ⎝⎠0 ⎩⎭⎪⎪⎣⎦()ab− /2 ⎣⎦−+()ab/2 exp ⎡⎤ikr− ω t ⎧ ⎛⎞EL ⎣⎦()0 ⎪ ⎡⎤⎡⎤ik() a+− bsinθθ ik() a b sin =−⎜⎟ ⎨exp⎢⎥⎢⎥ exp ⎝⎠rik0 sinθ ⎩⎪ ⎣⎦⎣⎦ 2 2 ⎡⎤⎡⎤−−ik() a bsinθθ −+ ik() a b sin ⎪⎫ +−exp⎢⎥⎢⎥ exp ⎬ ⎣⎦⎣⎦22⎭⎪ bikrtexp ⎡⎤− ω ⎛⎞EL ⎣⎦()0 EiP = ⎜⎟ {}exp()α ⎣⎡ exp()iiββ−−+ exp ( )⎦⎣⎤⎡ exp ( − iii αββ ) exp () −− exp ( ) ⎦⎤ ⎝⎠ri0 2 β whereαθβθ== k asin and k b sin But we know that exp()iiααα+−= exp ( ) 2 cos exp()iiiβββ−−= exp ( ) 2 sin Substituting we obtain bikrtexp ⎡⎤−ω ⎛⎞EL ⎣⎦()0 EiP = ⎜⎟ ()()2cosαβ 2 sin ⎝⎠ri0 2 β The irradiance at point P is given by 2 2 11*2⎛⎞EbL ⎛⎞ 4sinβ I=εε00 cEPP E= c⎜⎟()4cos α⎜⎟2 22⎝⎠r0 ⎝⎠ 4β 2 2 2 ⎛⎞sinβ 1 ⎛⎞EL b I=4cos I000αε⎜⎟2 , whereI= c⎜⎟ ⎝⎠β 2 ⎝⎠r0 FraunhoferFraunhofer Diffraction Diffraction fromfrom aa DoubleDouble SlitSlit The irradiance at point P from a double slit is given by the product of the diffraction pattern from single slit and the interference pattern from a double slit 2 2 ⎛⎞sin β II= 4cos0 α ⎜⎟2 ⎝⎠β FraunhoferFraunhofer Diffraction Diffraction fromfrom aa DoubleDouble SlitSlit Single Slit Double Slit 11-6.11-6. FraunhoferFraunhofer Diffraction Diffraction fromfrom ManyMany SlitsSlits (Grating)(Grating) Now for the multiple slits we just need to again change the limits of integration. For N even slits with width b evenly spaced a distance a apart, we can place the origin of the coordinate system at the center obstruction and label the slits with the index j (Note that the diagram does not exactly correspond with this). jN= /2 ⎡⎤()21jab−+ /2 ⎛⎞EL ⎣⎦ EikrtiksdsP =−⎜⎟exp⎣⎦⎡⎤()0 ωθ∑ exp ( sin ) {∫⎣⎦⎡⎤()21jab−− /2 ⎝⎠r0 j=1 ⎣⎦⎡⎤−−+()21jab /2 + exp()iks sinθ ds ∫⎣⎦⎡⎤−−−()21jab /2 } Integrating we obtain ⎡⎤21jab−+ /2 jN= /2⎧ ⎣⎦() ⎛⎞EL ⎪⎡⎤exp()iks sinθ EikrtP =−⎜⎟exp ⎣⎦⎡⎤()0 ω ∑ ⎨⎢⎥ ⎝⎠rik0 j=1 ⎣⎦sinθ ⎩⎪ ⎣⎦⎡⎤()21jab−− /2 ⎣⎦⎡⎤−−+()21jab /2 ⎫ ⎡⎤exp()iks sinθ ⎪ + ⎢⎥⎬ ⎣⎦iksinθ ⎣⎦⎡⎤−−−()21jab /2⎭⎪ exp⎡⎤ikr−−+−−ω tjN= /2⎧ ⎡ ik ⎡ 2 j 1 a b ⎤ sinθθ⎤⎡ ik ⎡ 2 j 1 a b ⎤ sin ⎤ ⎛⎞EL ⎣⎦()0 ⎪ ⎣() ⎦ ⎣() ⎦ =−⎜⎟ ⎨exp⎢ ⎥⎢ exp ⎥ riksinθ ∑ 2 2 ⎝⎠0 j=1 ⎩⎪ ⎣⎢ ⎦⎣⎥⎢ ⎦⎥ ⎡⎤⎡⎤−−−ik⎣⎦⎡⎤()21 j a b sinθθ −−− ik ⎣⎦⎡⎤() 21 j a b sin⎪⎫ +−exp⎢⎥⎢⎥ exp ⎬ 22 ⎣⎦⎣⎦⎢⎥⎢⎥⎭⎪ Substuting using α ==k asinθβ and k b sin θ and rearranging we obtain bikrtexp ⎡⎤−ω jN= /2 ⎛⎞EL ⎣⎦()0 EijiiijiiP =−−−+−−−−⎜⎟ ∑ {exp⎣⎡ () 2 1αβ⎦⎣⎤⎡ exp ()() exp β ⎦⎤ exp() () 2 1 αβ ⎣⎡ exp() exp ( β ) ⎦⎤} ⎝⎠ri0 2 β j=1 We can rewrite this as bikrtexp ⎡ −ω ⎤ jN= /2 ⎛⎞EL ⎣ ()0 ⎦ EP = ⎜⎟ ()2siniijijβαα∑ {} exp⎣⎦⎣⎦⎡⎤⎡⎤ () 2−+−− 1 exp () 2 1 ⎝⎠r0 2i β j=1 jN= /2 ⎛⎞EL ⎛⎞sin β =−⎜⎟bikrtexp⎣⎦⎡⎤()0 ωα⎜⎟∑ Re{} exp ⎣⎦⎡⎤ ij() 2 − 1 ⎝⎠r0 ⎝⎠β j=1 jN= /2 ⎛⎞EL ⎛⎞sin β =−⎜⎟bikrtexp⎣⎦⎡⎤()0 ωαααα⎜⎟∑ Re{} exp() i ++++− exp ( i 3 ) exp ( i 5 )L exp ⎣⎦⎡⎤ iN ( 1 ) ⎝⎠r0 ⎝⎠β j=1 The last term is a geometric series that converges to jN= /2 sin Nα ∑ Re{} exp()iiααα++++−= exp ( 3 ) exp ( i 5 )L exp⎣⎦⎡⎤ iN ( 1 ) α j=1 sinα The details of the last step are outlined in the book.int The irradiance at po P is given by 2 2 2 2 2 ⎛⎞sinβ ⎛⎞ sin Nα 11* ⎛⎞EbL ⎛⎞ sinβ ⎛⎞sin Nα IcEEcPPP==εε00⎜⎟⎜⎟⎜⎟ II= 22r β α P 0 ⎜⎟⎜⎟ ⎝⎠0 ⎝⎠⎝⎠ ⎝⎠βα⎝⎠sin FraunhoferFraunhofer Diffraction Diffraction fromfrom MultipleMultiple SlitsSlits The irradiance at point P is given by 2 2 ⎛ sin β ⎞ ⎛ sin Nα ⎞ 2 2222 ⎜ ⎟ ⎛⎞ ⎜ ⎟ 1* 1EbL ⎛⎞ sinβα⎛⎞ sinNN ⎛⎞ sin βα ⎛⎞ sin β ⎝ sinα ⎠ IcEEcPPP==εε00⎜⎟⎜⎟⎜⎟ = I 0 ⎜⎟ ⎜⎟ ⎝ ⎠ 22⎝⎠r0 ⎝⎠βα⎝⎠ sinsin ⎝⎠ βα ⎝⎠ N2 sin Nα Whenαπ= m, the term is a maximum.
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