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Course outline 1. Maxwell Eqs., EM waves, wave-packets 2. Gaussian beams 3. Fourier optics, the lens, resolution 4. Geometrical optics, Snell’s law 5. Light-tissue interaction: scattering, absorption Fluorescence, photo dynamic therapy חבילות גלים, גלים אלקטרומגנטים, משואות מקסוול.1 קרניים גאוסיניות.2 הפרדה, העדשה, אופטיקת פורייה.3 חוק סנל, אופטיקה גיאומטרית.4 , פלואורסנציה, בליעה, פיזור:רקמה- אינטראקציה אור.5 דינמי-טיפול פוטו 6. Fundamentals of lasers עקרונות לייזרים.6 7. Lasers in medicine לייזרים ברפואה.7 8. Basics of light detection, cameras 9. Microscopy, contrast mechanism 10.Confocal microscopy מצלמות, עקרונות גילוי אור.8 ניגודיות, מיקרוסקופיה.9 מיקרוסקופיה קונפוקלית.10 Fourier optics and imaging • Linear optical systems • Fresnel diffraction • Fraunhofer diffraction • The lens • Optical resolution Light propagation - Intuition Light from a point source propagates in spherical waves: CCD array Point source The action of a lens CCD array spot size 1.22 clipping resolution drop Point source f D a Van-leeuwenhoek microscope Conventional microscopy Can light interact with itself ? Fourier optics An arbitrary wave in free space may be analyzed as a superposition of plane waves. If it is known how a linear optical system modifies plane waves, the principle of superposition can be used to determine the effect of the system on any arbitrary wave. The linear-systems approach The complex amplitudes in two planes normal to the optical (z) axis are regarded as the input and output of the system. A linear system may be characterized by: - its impulse response function - the response of the system to an impulse (a point) at the input. - its transfer function - the response to spatial harmonic functions. f x, y U x, y , 0 g x, y U x, y , d Spatial harmonics An arbitrary function f(t) may be analyzed as a sum of harmonic functions of different frequencies and complex amplitudes. f t F ei 2 t d In two (spatial) dimensions:* f x, y F x , y e i 2 x x y y * Our definitions of temporal and spatial Fourier transforms differ in the sign of the exponent. The choice of this signs is arbitrary, as long as opposite signs are used in the Fourier and inverse Fourier transforms. With different signs in the spatial (2D) and temporal (1D) cases, the traveling wave exp[i(2t-kzz)] represents wave that moves in +z direction as time propagates. A little on Fourier transform Example 1 f(x,y) |F(x,y)|2 Example 2 f(x,y) |F(x,y)|2 Log (|F(x,y)|2) Example 3 f(x,y) |F(x,y)|2 Log (|F(x,y)|2) P(x,y) Example 4 f(x,y) |F(x,y)|2 Log (|F(x,y)|2) Spatial harmonics & Plane waves Consider a plane wave: (t=0, monochromatic) with a wavevector U x, y, z Ae i kx x k y y k z z A: Complex amplitude k kx , k y , kz k k k x2 k y2 k z2 2 . This wave propagates at angles: x sin 1 k x k y sin 1 k y k x = 0 means that there is no component of k in the x-axis (kx=0). At z=0: We also know (previous slides) that U is comprised of spatial harmonics: U x, y, 0 Ae i kx x k y y U x, y, 0 F x , y e i 2 x x y y Spatial harmonics & Plane waves A single plane wave: U x, y, 0 Ae i kx x k y y A single spatial harmonic: U x, y, 0 Fe i 2 x x y y x k x 2 y k y 2 (!) x , y k x , y 2 → Spatial frequency [cycles/mm] kc 2 → Optical frequency [cycles/sec] x sin 1 k x k y sin 1 k y k x sin 1 x y sin 1 y The angles of inclination of the wavevector are then directly proportional to the spatial frequencies of the corresponding harmonic function. Apparently, there is a one-to-one correspondence between the plane wave U(x,y,z) and the harmonic function f(x,y). Spatial harmonics & Plane waves plane wave U(x,y,z) harmonic function f(x,y) Given one, the other can be readily determined, provided the wavelength is known: - The harmonic function f(x,y) is obtained by sampling at the z0 plane, f(x,y) = U(x,y,z0). - Given the harmonic function f(x,y), on the other hand, the wave U(x,y,z) is constructed by using the relation: U x, y, z f x, y eikz z With: k z k 2 k x2 k y2 k x sin 1 k x k sin 1 x 2 y sin 1 k y k sin 1 y A condition for the validity of this correspondence is that kz is real ( kx k y k ). This condition implies that x < 1 and y < 1, so that the angles x and y exist. 2 2 2 The signs represent waves traveling in the forward and backward directions. We shall be concerned with forward waves only. Spatial-spectral analysis With a single spatial frequency: U x, y , z f x , y e ik z z U x , y , 0 f x, y e e i 2 x x y y i 2 x x y y e ikz z If the transmittance of the optical element f(x,y) is the sum of several harmonic functions of different spatial frequencies, the transmitted optical wave is also the sum of an equal number of plane waves dispersed into different directions. The amplitude of each wave is proportional to the amplitude of the corresponding harmonic component of f(x,y). Spatial-spectral analysis Mathematically, if f(x,y) is a superposition integral of harmonic functions, f x, y F x , y e i 2 x x y y d x d y with frequencies x and y, and amplitudes F(x,y), the transmitted wave U(x,y,z) is the superposition of plane waves, U x, y, z F x , y e i 2 x x y y eikz z d x d y with complex envelopes F(x ,y) and k z For any z: k 2 k x2 k y2 =2 2 x2 y2 . f x, y F , e i 2 x x y y d d x y x y F x , y f x, y ei 2 x x y y dxdy Spatial-spectral analysis A thin optical element of complex amplitude transmittance f(x,y) decomposes an incident plane wave into many plane waves. Each wave travels at angles x = sin-1(x) and y = sin-1(y) and has a complex envelope F(x,y). This process of "spatial spectral analysis" is analogous to the angular dispersion of different temporal-frequency components (wavelengths) by a prism. Free-space propagation serves as "spatial prism“, sensitive to the spatial rather than temporal frequencies of the waves. The transfer function input plane f x, y U x, y,0 output plane g x, y U x, y, d We regard f(x,y) and g(x,y) as the input and output of a linear system. The system is linear since the Helmholtz equation, which U(x,y,z) must satisfy, is linear. Linear systems Shift-invariant system: Invariance of free space to displacement of the coordinate system. A linear shift-invariant system is characterized by its impulse response function h(x,y) or by its transfer function H(x,y). Impulse response Transfer function Transfer function of free space consider a single harmonic input function, i 2 x x y y f x, y F x , y e which corresponds to a plane wave: d x d y A e U x, y, z A e i 2 x x y y i kx x k y y kz z where: k z k 2 k x2 k y2 =2 2 x2 y2 x k x 2 y k y 2 After propagating a distance d: g x, y A e i kx x k y y kz d Thus transfer function of free space: g x, y i 2 d ik z d H x , y e e f x, y 2 x2 y2 Transfer function of free space The (complex) transfer function of free space: H x , y e i 2 d 2 x2 y2 sphere Fourier optics and imaging • Linear optical systems • Fresnel diffraction • Fraunhofer diffraction • The lens • Optical resolution Fresnel approximation H x , y e i 2 d 2 x2 y2 If the input function f(x,y) contains only spatial frequencies that are much smaller than the cutoff frequency -1, so that 2 2 2 x y , the plane wave components of the propagating light then make small angles xx and yy , corresponding to paraxial rays. x2 y2 2 2 d x y d i 2 1 i 2 1 2 2 2 2 i 2 d x y 2 2 * x y H , H x , y * e e e 2 2 ikd i d x y e e 1 1 2 parabola sphere Input-output relation (Fresnel) Given the input function f(x,y), the output function g(x,y) may be determined as follows: 1. we determine the Fourier transform F x , y f x, y e i 2 x x y y dxdy 2. the product H(x,y) F(x,y) gives the complex envelopes of the plane wave components in the output plane. 3. the complex amplitude in the output plane is the sum of the contributions of these plane waves: i 2 x x y y g x, y H x , y F x , y e d x d y H x , y e Using the Fresnel approximation for H(x,y), we have: g x, y e ikd F x , y e i d x2 y2 e i 2 x x y y 2 2 ikd i d x y e d x d y Impulse response of free space The impulse response function h(x,y) of the system of free-space propagation is the response g(x,y) when the input f(x,y) is a point at the origin (0,0). F x , y f x, y e i 2 x x y y dxdy x, y e i 2 x x y y g x, y 1 H x , y e which is the inverse Fourier transform of H x , y e i h x, y g x, y e d i 2 x x y y 2 2 ikd i d x y e x2 y 2 ik ikd 2d e Thus, each point in the input plane generates a paraboloidal wave; all such waves are superimposed at the output plane. d x d y (exercise): dxdy 1 Free space propagation as a convolution (Fresnel) Knowing h, we can regard f(x,y) as a superposition of different points (delta functions), each producing a paraboloidal wave. The wave originating at the point (x’,y’) has an amplitude f(x’,y’) and is centered about (x’,y’) so that it generates a wave with amplitude f(x’,y’)h(x-x’,y-y’) at the point (x,y) in the output plane. The sum of these contributions is the two-dimensional convolution: g x, y f x, y h x x, y y dxdy, which in the Fresnel approximation h x, y i e d i ikd g x, y e f x, y e d x2 y 2 ik ikd 2d e becomes: 2 2 x x y y i d dxdy. Unlike previous derivation of g from f, here no Fourier transform is involved. Convolution f h f g t d Fresnel approximation: summary Within the Fresnel approximation, there are two approaches for determining the complex amplitude g(x,y) in the output plane, given the complex amplitude f(x,y) in the input plane: 1. space-domain approach in which the input wave is expanded in terms of paraboloidal elementary waves. i ikd g x, y e f x, y e d 2 2 x x y y i d dxdy 2. frequency-domain approach in which the input wave is expanded as a sum of plane waves: g x, y e ikd F x , y e i d x2 y2 e i 2 x x y y d x d y Fourier optics and imaging • Linear optical systems • Fresnel diffraction • Fraunhofer diffraction • The lens • Optical resolution Fraunhofer approximation Start with the space-domain approach of Fresnel approximation: i ikd g x, y e f x, y e d i ikd e f x, y e d 2 2 x x y y i d x i 2 dxdy y 2 x2 y 2 2 xx yy d dxdy If f(x,y) is confined to a small area of radius b, and if the distance d is sufficiently large so 2 2 that b2/d is small, then the phase factor x y d is negligible: i g x, y e d i e d x2 y 2 i ikd d e f x, y e i 2 xx yy d dxdy x x d y y d x2 y 2 i ikd d e f x, y e i ikd i x dy x y e e F , d d d 2 2 i 2 x x y y dxdy x sin 1 x sin x x sin x tan x x x x x d d Fraunhofer approximation x y g x, y h0 F , d d i h0 e d x2 y 2 i ikd d e In the Fraunhofer approximation, the complex amplitude g(x,y) of a wave of wavelength in the z=d plane is proportional to the Fourier transform F(x,y) of the complex amplitude f(x,y) in the z=0 plane, evaluated at the spatial frequencies x =x/d and y =y/d. The approximation is valid if f(x,y) at the input plane is confined to a circle 2 of radius b satisfying: 2 x d y max d f x, y 2 b2 “Fraunhofer region” 2 g x, y Fresnel region 2b Fraunhofer region Fraunhofer – FT in the far-field One slit. The width of the silts is varied. Two slits. The width of the silts is constant and the distance between them is varied. Periodic objects - gratings f x 1 cos 2 Gx F x f x, y e i 2 x x y y dxdy 1/G G: lines per meter i 2 x 1 cos 2 Gx e x dx g x 1 1 x x G x G 2 2 x x 1 x 1 x F G G d d 2 d 2 d g x 2 1 x y 2 2 F , d d d 1 2d 2 b2 2 d 2b f x 2 x 1 x 1 x G G d 4 d 4 d g x 2 1/(2d2) 1/(42d2) -dG 0 dG x Sarcomere contractions Sarcomeres are multi-protein complexes composed of different filament systems. http://highered.mcgraw-hill.com/sites/0072495855/student_view0/chapter10/animation__sarcomere_contraction.html Sarcomere contractions -dG dG x d Helium-Neon laser = 632 nm Fourier optics and imaging • Linear optical systems • Fresnel diffraction • Fraunhofer diffraction • The lens • Optical resolution Angular spectrum - definition Definition: The angular spectrum A(x,y,z) of a wave U(x,y,z) emerging from an object: AU FT pair: A , , z U x, y, z ei 2 x x y y dxdy x y U x, y, z A x , y , z ei 2 x x y y d x d y Thus A(x,y,z) is simply the equivalent for the Fourier transform F of the object f(x,y): Angular spectrum Fourier transform of the object A x , y , z F x , y eikz z U x, y, z f x, y e ikz z Complex wave Object complex transmission Propagation of angular spectrum Substitute U A x , y , z ei 2 x y d x d y into Helmholtz equation 2U x, y, z k 2U x, y, z 0 x y And executing the derivatives of the x and y coordinates gives: d2 i 2 x x y y 2 2 2 2 A , , z k 4 A , , z d x d y 0 x y x y e dz 2 x y =0 Which yields a differential equation: d2A 2 2 2 2 k 4 x y A0 2 dz with a solution: A x , y , z A x , y , 0 e iz k 2 4 2 x2 y2 2 2 2 Fresnel approximation: x y 1 1 2 A x , y , 0 e A x , y , 0 e izk 1 2 x2 y2 1 izk 1 2 x2 y2 2 Propagation of the angular spectrum (Fresnel approximation) A x , y , z A x , y ,0 e 2 2 ikz i z x y e Phase transformation with a thin lens From straight-forward geometrical considerations: R1 x2 y 2 x2 y 2 x, y t0 R1 1 1 R2 1 1 2 2 R R Lens thickness 1 2 R2 t0 Paraxial approximation R >> lens diameter: n Lensmaker’s equation for a thin lens in air: 1 1 f n 1 R1 R2 1 The pupil function: P(x,y) = 1, inside the aperture 0, otherwise x2 y 2 x2 y 2 1 1 2 R 2R2 x2 y 2 1 1 x, y t0 2 R1 R2 The phase transformation (x,y) by the lens is given by (k=2/): x, y kn x, y k t0 x, y kt0 k n 1 x, y glass x, y knt0 air k x2 y 2 2f Therefore, the phase transformation of a perfect thin lens is: t x, y e iknt0 i e k 2 2 x y 2f Fraunhofer diffraction by a lens The field after a (thin) lens (neglecting the exp(iknt0) term): U x, y , 0 U x, y , 0 P x, y e Using the Fresnel integral: U x, y , z i ikz e U x, y, 0 e z 2 2 x x y y i z i k x2 y 2 2f U x, y , 0 U x, y , 0 f U x, y, f dxdy 2 k i x2 y 2 i xx yy i ikz i 2kz x2 y 2 z 2z e e U x , y , 0 e e dxdy z i x i ikz i 2kz x2 y 2 2f e e U x, y, 0 P x, y e z k U x, y , f i f i f e ikf e i k 2 2 x y 2f 2 y 2 2 xx yy f e U x , y , 0 P x, y e i U x , y , 0 P x, y e i 2 x x y y i k x 2 y 2 2z e e ikf e k 2 2 x y 2f 2 xx yy z dxdy dxdy i i dxdy Lens Fraunhofer diffraction of U(x,y,0-) multiplied by the pupil function x x f y y f Fourier transform with a lens U x, y, f ie ikf f e k 2 2 i x y 2f U x , y ,0 P x, y e i 2 x x y y U x, y , 0 U x, y , 0 dxdy d1 f Assume U(x,y,0-) has an extent less than P(x,y): i x ie ikf 2f U x, y , f A x , y , 0 e f k 2 y2 Propagation of the angular spectrum from -d1 to 0-: A x , y ,0 A , U x, y, f x ie y , d1 e ik f d1 f 2 2 ikd1 i d1 x y e U x, y, d1 U x, y, f Reminders: A x , y , z U x, y, z e i 2 x x y y A x , y , z A x , y ,0 e A x , y , d1 e i k d1 2 2 1 x y 2f f dxdy 2 2 ikz i z x y e x f y y f x d1 f iei 2 kf U x, y, f A x , y , f f The field at the focal plane of the lens is the 2D Fourier transform of the field at z = -f Fourier transform with a lens iei 2 kf U x, y, f A x , y , f f Fourier transform property of a lens: A x , y , z F x , y eikz z U x, y, z f x, y e ikz z x y iei 2 kf g x, y F , f f f The complex amplitude of light at a point (x,y) in the back focal plane of a lens of focal length f is proportional to the Fourier transform of the complex amplitude in the front focal plane evaluated at the frequencies x/λf, y/λf. This relation is valid in the Fresnel approximation. Without the lens, the Fourier transformation is obtained only in the Fraunhofer approximation, which is more restrictive. Image formation with a lens U x, y , 0 Assume a positive, aberration-free thin lens and monochromatic light. Free space propagation as a convolution (Fresnel): g x, y f x, y h x x, y y dxdy ieikd d f x, y e i x x 2 y y 2 d U x, y , 0 x1 , y1 d1 dx dy di U x, y, d1 U xi , yi , di To find h, we replace f(x’,y’) U(x,y,-d1) (x-x1,y-y1,-d1): U x, y , 0 U x, y , 0 ie ikz z x x1 , y y1 , d1 e 2 2 x x y y i z dxdy e U x, y , 0 P x, y e i k 2 2 x y 2f U xi , yi , di h xi , yi , x1 , y1 U x, y , 0 e i k 2 di x x y y 2 i 2 i dxdy P x, y e k 1 1 1 i x2 y 2 2 d1 di f e x x y y ik i 1 x i 1 y di d1 di d1 dxdy 2 2 x x1 y y1 ik 2 d1 Image formation with a lens h P x, y e k 1 1 1 i x2 y 2 2 d1 di f =0 e x x y y ik i 1 x i 1 y di d1 di d1 1 1 1 di f d1 dxdy h xi , yi , x1 , y1 P x, y e d1 The lens law 2 x Mx1 x yi My1 y di i e di U x, y, d1 U xi , yi , di M dxdy Neglect P(x,y): U x, y , 0 x1 , y1 In this case, the impulse response becomes i U x, y , 0 di d1 Magnification x i 2 x1 i M yi x y1 M y dxdy xi yi x1 , y1 M M Imaging by a lens: y x U xi , yi , di U 1 , 1 , d1 M M ei 2 xx dx x • Inversion • Magnification Imaging examples The lens law U x, y f f f f x y U , 4 4 U x, y d1 5 f 4 F f f f U x, y F f x y A , f f U x, y F f f f1 f1 f M di d1 M f2 f1 U x, y f f x y A , f f f1 f Magnification F di 5 f x y U , 2 2 f2=2f f2 f2 F F F d1 2 f 1 1 1 di f d1 F di 2 f U x, y The 4-f system Imaging examples A x , y U x, y f Object f f Mask U x, y f Image (inverted) Fourier optics and imaging • Linear optical systems • Fresnel diffraction • Fraunhofer diffraction • The lens • Optical resolution Point-spread function (PSF) PSF ↔ Impulse response Diffraction limited (also “Fourier limited”) system: Perfect spherical wave Point object Image Point-spread function (PSF) U x, y , 0 PSF ↔ Impulse response h xi , yi , x1 , y1 P x, y e k 1 1 1 i x2 y 2 2 d1 di f e x x y y ik i 1 x i 1 y di d1 di d1 dxdy d1 di U x, y, d1 U xi , yi , di Consider: • An impulse at x1=y1=0 • An imaging condition: x1 , y1 0, 0 U x, y , 0 1 1 1 d1 d i f h xi , yi P x, y e xx y y i 2 i i di di P di x, di y e Circular pupil function: (D - lens diameter) 1, x 2 y 2 D 2 P x, y 0, otherwise x2 y 2 1, D 2 P 0, otherwise dxdy x’=x/λdi (neglect constants) i 2 xi x yi y dxdy 2D Fourier transform of the scaled pupil PSF FT P J1 2 D 2 J1 d i h x, y h 0, 0 D di s 1.22 di h di D 2 yi d s 1.22 i D “Airy disk” xi Airy disk of microscope objectives D D NA n sin n sin arctan n 2f 2f f 1 In photography: “angular aperture” or “f-number” F# D 2 NA D f n θ n 1 32 48 58 NA 0.6 NA 0.8 NA 1.3 Reminder Gaussian beams - properties Beam divergence z z W z W0 1 z0 W z0 W z 0 z z0 NA for Gaussian beams 2 W0 U r A0 e W z W0 W0 0 2 NA n sin 0 n z0 W0 W0 W0 4 Thus the total angle is given by 2 0 2W0 2 W 2 z e z W z W0 1 z0 ikz ik 2 2 R z 2 z0 2 R z z 1 z z tan 1 W0 z z0 z0 2W0 2 0 4 i z Resolution The final image is a convolution of the perfect image with the system’s impulse response: y x U xi , yi , di h xi , yi U i , i , d1 M M Convolution f h f g t d 2D convolution examples Quantifying resolution The Rayleigh criterion Rayleigh criterion “Two point sources are just resolved if they have an angular separation equal to the angular radius of the Airy disk.” 1 sin r 1.22 D f For an ideal lens: l 1.22 D 1 For a microscope objective: l 0.61 NA NA n sin n D 2f Effect of noise on resolution Summary An arbitrary wave may be analyzed as a superposition of plane waves. U(x,y,0)=f(x,y)e-ikz can be represented as a combination of spatial harmonics: Fresnel approximation ieikz U xi , yi , z z Propagation of the angular spectrum U x, y, 0 e i x x 2 yi y 2 z i A x , y , z A x , y ,0 e At its focal plane a lens performs a Fourier transform of the incoming field x y iei 2 kf g x, y F , f f f The 4-f system allows Fourier domain image manipulations s 1.22 di D 2 2 ikz i z x y ieikz i 2kz xi2 yi2 e FT U x, y, 0 Fraunhofer approximation U xi , yi , z z The PSF of a lens is limited by its pupil function dxdy e