Fresnel and Fraunhofer Diffraction 4. 1 Background 4.1.1 The intensity of a wave field 4.1.2 The Huygens-Fresnel principle in rectangular coordinates 4.2 The Fresnel approximation 4. 2. 1 Positive vs. negative phases 4.2.2 Accuracy of the Fresnel approximation 4.2.3 The Fresnel approximation and the angular spectrum 4.2.4 Fresnel diffraction between confocal spherical surfaces 4. 3 The Fraunhofer approximation 4.4 Examples of Fraunhofer diffraction patterns 4.4.1 Rectangular aperture 4.4.2 Circular aperture 4. 4. 3 Thin sinusoidal aperture grating 4.4.4 Thin sinusoidal phase grating 4.5 Examples of Fresnel diffraction calculations 4.5.1 Fresnel diffraction by a square aperture 4. 5 2 Fresnel diffraction by a sinusoidal amplitude grating-Talbot images 4.1 Background • In chapter 3 we dealt with most general form of the diffraction theory. • In c hap ter 4 we w ill dea l w ith – Intensity of a wave field – Huygens-Fresnel principle – Certain approximations to reduce the problem to a simpler mathematical form. These approximations are: • Fresnel • Fraunhofer – We consider the wave propagation phenomenon as a system. – The approximations will be valid for certain class of inputs. – Preparation for the calculations related to the approximations Spring 2010 Eradat Physics Dept. SJSU 2 4.1.1 The intensity of a wave field Intensity is the physically measurable attribute of an optical wavefield Intensity and power density are not the same but proportional Intensity of a scalar monochromatic wave at point P IP()|= UP ()|2 For a narrow-band (not perfectly monochromatic) intensity is given by IP( )== | uPt ( ,)|2 and IPt ( ,) | uPt ( ,)|2 An infinite Instantaneous time average Intensity In calculating a diffraction pattern, we are looking for the intensity of the pa ttern. Spring 2010 Eradat Physics Dept. SJSU 3 4.1.2 The Huygens-Fresnel principle in rectangular coordinates According to the first Rayleigh-Somefeld solution the diffracted field on the xy plane due to the aperture on the ξη plane is UI ( P0 ) and the Huygens-Fresnel principle can be written as: 1 e jkr01 z UP( ) = UP() cosθθdsnrwhere cos== cos()( , ) I 01∫∫Σ 01 jλ rr01 01 zejkr01 Uxy( , ) = U()(ξη, ) dd ξ η, ∫∫Σ 2 jrλ 01 η 222 y where rzx01 =+−+−()()ξη y ξ x Two approximations are used in this result: P0 1) inherent approximation in the scalar theory Σ z θ 2) r01 >> λ P Spring 2010 Eradat Physics Dept. SJSU 1 4 4.2 The Fresnel approximation I Goal: to reduce the H-F principle to a simple and usable expression WhithibWe achieve this by approx itifimations for r01 : nn(1)− Binomial expansion: (xy+=+ )nn x nxy n−−122 + x n y ++= ... yn n, 1,2,3,... 2! 11 (1+=±bbb )±1/2 1 ∓ 2 ±−<≤... for 1b 1: the higher order terms are negligable 28 22 2 2 ⎛⎞⎛⎞xy−−ξη⎡⎤11 ⎛⎞⎛⎞ x − ξ y − η rz01 =+11⎜⎟⎜⎟ + ≈+ z⎢⎥ ⎜⎟⎜⎟ + +... ⎝⎠⎝⎠zz⎣⎦⎢⎥22 ⎝⎠⎝⎠ z z Where do we cut the series? We will use r01 in the diffracted field equation zejkr01 Ux( ,(,)y) = Uddξ ηξη ∫∫Σ 2 jrλ 01 jkr01 The term eris very sensitive to the values of 01 specially since it is multiplied blby a very large num ber k = 2/πλλ. IthiiblIn the visible ofthf the or der 107. We keep two 2 terms for the exponent. For rz01 error introduced by dropping all terms but is small. ⎧⎫k jkr01 jkz ∞ ⎡⎤22 zee ⎨⎬jx⎣()()−+−ξη y⎦ UUddUU ( x, yU) ==()(ξξξ,η)( ddUξ η () ξ,η)e⎩⎭2z dd ξη ∫∫Σ 2 ∫ ∫ jrjzλλ01 −∞ The integration limit is let to ∞ using the usual boundary conditions. Spring 2010 Eradat Physics Dept. SJSU 5 4.2 The Fresnel approximation II jkz ∞ ⎧⎫k ⎡ 22⎤ e ⎨⎬jx⎣⎦()()−+−ξη y Uxy(),(,)= ∫∫ Uξη e⎩⎭2z dd ξ η this looks like a convolution jzλ −∞ ∞ jkz ⎧⎫k ⎡⎤22 e ⎨⎬jx⎣⎦()()−+ξη+ y− Uxy( ,(,)(,)(,)) =−−= Uξη hx ξ y η dd ξ η where hxy e⎩⎭2z ∫∫ jzλ −∞ First form of the Fresnel diffraction integral Another form of the Fresnel diffraction integral is expressed as the following k j ξη22+ Fourier transform of the U (ξ ,)η e 2 z( ) which is complex field just to the right of aperture multiplied by a quadratic phase factor jkz kk22∞ 22 2π e jxy()++⎧⎫ j()ξη −+jxy()ξη Uxy(),(,)= e22zz⎨⎬ Uξη e eλz dd ξ η jzλ ∫∫ −∞ ⎩⎭ Seond form of the Fresnel diffraction integral Observation in the near field of the aperture or Fresnel diffraction region xy--ξη Where r >>λ,1,1, < < 01 zz and scalar theory approximation are assumed Spring 2010 Eradat Physics Dept. SJSU 6 4.2.1 Positive vs. negative phases Goal: to understand the meaning of the signs of the y phase exponentials: in the sppqherical wave e jkr01 and its equivalent k jxy()22+ in the quadratic approximation ez2z (0)for > z Sign convention: our phasors rotate in the clockwise direction (the angle becomes more negative as time goes) and their time dependence is e− jt2πν We move in space in such a way that we encounter Wavefront Wavefront emitted emitted portions of the wavefield that were emitted earlier in time. later earlier The phase must become more positive since the y phasor had not have time to rotate as far in clockwise. k We move in space in such a way that we encounter θ z portions of the wavefield that were emitted later in time. The phasor will have advanced in the clockwise direction, therefore the phase must become more negative. Spring 2010 Eradat Physics Dept. SJSU 7 4.2.2 Accuracy of the Fresnel approximation I Fresnel approximation replaced the spherical secondary wavelets with parabolic wavefronts in the Huygens-Fresnel principle jkr jkz ∞ ⎧⎫k 2 2 01 ⎨⎬jx⎡⎤()()−+−ξη y ze e ⎩⎭2z ⎣⎦ Uxy( , ) = U()(ξη, )(2 dd ξ η→ U () ξη, ) e ddξη jrjzλλ∫∫Σ ∫∫ 01 −∞ Parabolic wavelets Spherical wavelets The accuracy of this approximation depends on the size of the higher order terms in binomial expansion. A sufficient condition for accuracy is: ⎡⎤ ⎢⎥ 22 222 ⎢⎥1 ⎛⎞⎛⎞x −ξ 11yxy−−−ηξη⎪⎪⎧⎫ ⎛⎞⎛⎞ rz01 ≈+⎢⎥1 ⎜⎟⎜⎟++⎨⎬ ⎜⎟⎜⎟ + ... ⎢⎥2 ⎝⎠⎝⎠z 28zzz⎩⎭⎪⎪ ⎝⎠⎝⎠ ⎢⎥ maximum phase change due to dropping ⎢⎥2 ⎣⎦b /8 term must be much less than one radian 2 φ ⎧⎫2 2 jkr 2 21πξ⎪⎪⎛⎞x − ⎛⎞y −η eO01 →Δφ() = ⎨⎬⎜⎟+<<⎜⎟ 1 λ 8 ⎝⎠z ⎝⎠z ⎩⎭⎪⎪Max π 222 zxy3 >>()() −ξη + − 4λ {}Max Spring 2010 Eradat Physics Dept. SJSU 8 4.2.2 Accuracy of the Fresnel approximation II Example: calculate the safe distance to use the Fresnel approximation for a circular aperture of size 1cm and a circular observation ragion of 10.525cm with a light of λμ=>> m. (Answere: z cm) π 222 zx3 >>()() −ξη+−yxy hint: − ξη and − should have their 4λ { }Max maximum possible values to evaluate the condition. If the higher order terms do not change the value of the Fresnel integral substantially, we can use the approximation. They do not need to be small in this case. Spring 2010 Eradat Physics Dept. SJSU 9 4.2.3 The Fresnel approximation and the angular spectrum I Goal: understand the implications of the Fresnel approximations from the point of view of angular spectrum method of analysis. We compare the transfer function of propagation through free space, predicted by RS scalar diffraction theory, with the transfer function predicted by the Fresnel analysis General spatial phase dispersion ⎧representing propagation ⎪ ⎪ ⎛⎞2π z 2 2 Hff(,)= ⎜⎟jff1()−−()λλXY ← RS theory RS X Y ⎨ ef⎝⎠λ 22+<1f / λ ⎪ X Y ⎩⎪0 otherwise A constant Qadratic phase phase delay dispersion due to traveling jkz ⎧⎫k ⎡⎤22 ⎨⎬jx()()−+−ξη y jzf22 f e ⎩⎭2z ⎣⎦FT jkz −+πλ ()XY hx(,yyff )=⎯⎯→=eHFXY (ff , ) e e jzλ All plane waves Different plane-wave suffer equally components suffer Fresnel approximation impulse response different phase delays Spring 2010 Eradat Physics Dept. SJSU 10 4.2.3 The Fresnel approximation and the angular spectrum II ⎛⎞2π z 2 2 ⎧ ⎜⎟jff1()−−()λλXY ⎪ ⎝⎠λ 22 eff XY+<1/ λ HffRS(,) X Y =←⎨ RS theory ⎩⎪0 otherwise 22 jkz −+jzfπλ ()XY f HffFXY(,)= ee We can see that HffFXY(,) is an approximation to the H RSXY (,) ff Applying the binomial expansion to the H RS(,)ff X Y we get: 22 222()λλffXY() 1()1−−≈−−()λλf f2 if () λ f 1 and () λ f 1 XY22 X Y ⎛⎞⎡⎤22 2π z ()λλffXY() ⎛⎞2π z 2 2 ⎜⎟j ⎢⎥1−− ⎜⎟jff1()−−()λλXY ⎜⎟λ ⎢⎥22 jzf22 f ⎝⎠⎝⎠λ ⎣⎦jkz −+πλ ()XY ee≈== eeHffFXY(,) Conclusion: 2 2 2 ⎛⎞kx HffRS(,) X Y≈== Hff F (,) X Y When the conditions: ()λα f X ⎜⎟ 1, ⎝⎠||k 2 2 2 ⎛⎞kY ()λβfY ==⎜⎟1 are satisfied. So Fresnel approxilation is equivalent ⎝⎠||k to the paraxial approximation that is limitted to small propagation angles. Spring 2010 Eradat Physics Dept. SJSU 11 4.2.4 Fresnel diffraction between confocal spherical surfaces I Goal: analysis of diffraction between two confocal spherical surfaces Confocal spheres: center of each lies on the surface of the other. We set the spheres tangant to the plance we used before. Located at zzzr==0. and 01 is the distance between two spherical caps. We write the equations for both surfaces and find the distance between them. Make paraxial approximation by using the binomial expansion. Assuming the extend of the spherical caps about the z axis is small compared to the radii of the spheres, i.e. for z>> x − ξ and y −η , we get xyξη η rzx=+−+−→≈222()()ξη y rz -- y 01 01 zz ξ x zejkr01 Uxy(),(,)= Uξη dξηd , ∫∫Σ 2 jλ r01 r01 jkz ∞ ⎧⎫2π e ⎨−+jxy[ ξη]⎬ Uxy,(,)= Uξ ηξη e⎩⎭λz dd ( ) ∫∫ jzλ −∞ Field on the right hand Fourier transform of the Field on the z spherical cap left hand spherical cap Spring 2010 Eradat Physics Dept.