EE 485, Winter 2004, Lih Y. Lin Chapter 4 Fourier Optics - Based on harmonic analysis (Fourier transform) and liner system (superposition). - An arbitrary function ∞ ∞ = ν ν − π ν +ν ν ν f (x, y) ∫∫F( x , y )exp[ j2 ( x x y y)]d xd y −∞−∞ → Superposition, or integral of harmonic functions of x and y. ν ν F( x , y ) : Complex amplitude ν ν x , y : Spatial frequency (cycles/unit length) - Compare this with plane wave = − + + U (x, y, z) Aexp[ j(kx x k y y kz z)] = − π ν +ν U (x, y,0) Aexp[ j2 ( x x y y)] k k ν ↔ x , ν ↔ y x 2π y 2π - An arbitrary function can be analyzed as a superposition of harmonic functions. → An arbitrary traveling wave U (x, y, z) may be analyzed as a sum of plane waves! 4.1 Propagation of Light in Free Space A. Correspondence Between the Spatial Harmonic Function and the Plane Wave 1 EE 485, Winter 2004, Lih Y. Lin −1k −1 θ = sin x = sin (λν ) x k x (4.1-1) − k − θ = sin 1 y = sin 1(λν ) y k y A physical way of picturing the spatial harmonic function is to project a plane wave on the x-y plane. Λ = 1 Λ = 1 x ν , y ν x y − − θ = 1 λ θ = 1 λ → x sin Λ , y sin Λ x y Paraxial approximation: θ = λ = λν θ = λ = λν x Λ x , y Λ y (4.1-2) x y Spatial spectral analysis (Response of a plane wave after a thin optical element.) Consider a simple case: = − π ν +ν t(x, y) exp[ j2 ( x x y y)] Λ = 1 Λ = 1 → Harmonic function on x-y plane with period x ν , y ν . x y = − π ν +ν − U (x, y, z) Aexp[ j2 ( x x y y)]exp( jkz z) θ = −1 λν θ = −1 λν → Output wave is bent with angles x sin ( x ), y sin ( y ). The harmonic function pattern works like a grating. Now consider a general case: = ν ν − π ν +ν ν ν t(x, y) ∫∫ F( x , y )exp[ j2 ( x x y y)]d xd y (4.1-4) 2 EE 485, Winter 2004, Lih Y. Lin = ν ν − π ν +ν − ν ν U (x, y, z) ∫∫ F( x , y )exp[ j2 ( x x y y)]exp( jkz z)d xd y 1 1 1 k = k 2 − k 2 − k 2 = 2π − − z x y λ2 λ2 λ2 x y → An incident plane wave is decomposed into many plane waves, each θ = −1 λν θ = −1 λν traveling at angles x sin ( x ), y sin ( y ), with a complex envelope ν ν F( x , y ) , the Fourier transform of f (x, y) . Example 4.1-2, Imaging πx2 t(x, y) = exp j = exp()− j2πϕ(x, y) λf x2 ϕ(x, y) = − 2λf ϕ ↔ν +ν Compare to earlier: (x, y) x x y y ∂ϕ(x, y) x Now ν varies with x ⇒ν = = − x x ∂x λf − x ⇒ θ = sin 1− x f → A cylindrical lens with focal length f 3 EE 485, Winter 2004, Lih Y. Lin B. Transfer Function of Free Space Since an arbitrary function can be analyzed as sum of harmonic functions, we consider a harmonic input function. = = − π ν +ν f (x, y) U (x, y,0) Aexp[ j2 ( x x y y)] = = − + + Output g(x, y) U (x, y,d) Aexp[ j(kx x k y y kz d)] g(x, y) Η(ν ,ν ) = = exp(− jk d) x y f (x, y) z 1 (4.1-6) 1 2 = exp− j2π −ν 2 −ν 2 d λ2 x y Fresnel approximation ν 2 +ν 2 << 1 x y λ2 → The plane-wave components of the propagating light make small angles θ λν θ λν x ~ x , y ~ y . → Paraxial waves: ν ν = − [ πλ ()ν 2 +ν 2 ] H ( x , y ) exp( jkd)exp j d x y (4.1-8) Validity of Fresnel approximation has the same expression as in Sec. 2.2. Input-output relation Given the input function f (x, y) , how to obtain the output g(x, y) : (1) Determine the complex envelopes of the plane-wave components in the input plane by Fourier transform. ∞ ∞ ν ν = π ν +ν F( x , y ) ∫∫f (x,y)exp[ j2 ( x x y y)]dxdy −∞−∞ 4 EE 485, Winter 2004, Lih Y. Lin (2) Complex envelopes of the plane-wave components in the output plane = ν ν ν ν H( x , y )F( x , y ) = ν ν ν ν − π ν +ν ν ν (3) g(x, y) ∫∫ H ( x , y )F( x , y )exp[ j2 ( x x y y)]d xd y Under Fresnel approximation, g(x, y) = H ∫∫ F(ν ,ν )exp[ jπλd(ν 2 +ν 2 )]exp[− j2π (ν x +ν y)]dν dν 0 x y x y x y x y ≡ − H 0 exp( jkd) Free-space propagation as a convolution Each point generates a spherical wave. Under Fresnel approximation (observation point close to the propagation axis), spherical wave → parabolic wave. x2 + y 2 h(x, y) ≈ h exp− jk (4.1-13) 0 2d j h = exp(− jkd) 0 λd (x − x')2 + (y − y')2 g(x, y) = h ∫∫ f (x', y')exp− jπ dx'dy' (4.1-14) 0 λd 4.2 Optical Fourier Transform A plane wave transmitting through an optical element can be used to ν ν decompose the harmonic functions (Fourier components F( x , y ) ) that compose the pattern ( f (x, y) ) on the optical element. A. Fourier Transform in the Far Field (Fraunhofer Approximation) If f (x, y) is confined to a small area of radius b, distance d to the observation 2 plane is sufficiently large, so that Fresnel number for f (x, y) , N '= b << 1. F λd 5 EE 485, Winter 2004, Lih Y. Lin x2 + y 2 x y g(x, y) = h exp− jπ F( , ) (4.2-4) 0 λd λd λd Furthermore, if we limit our interest to points at the output plane within a circle 2 of radius a centered about the z axis, so that N = a <<1 for g(x, y) . F λd x y g(x, y) = h F( , ) (4.2-1) 0 λd λd → The only plane wave that contributes to the complex amplitude at (x, y) at output plane is the wave making angles θ = x ,θ = y with the optical axis. x d y d This is also the wave with wave-vector components k = ()x k, k = y k and x d y d amplitude F(ν ,ν ) with ν = x ,ν = y . x y x λd y λd • Fraunhofer approximation is valid when both N F and N F ' are small. B. Fourier Transform Using a Lens 6 EE 485, Winter 2004, Lih Y. Lin θ θ = λν λν Amplitude of the plane wave with direction ( x , y ) ( x , y ) is ν ν proportional to the Fourier transform F( x , y ) and is located at the point = θ θ = λ ν λ ν (x, y) ( x f , y f ) ( f x , f y ) . x y → g(x, y) ∝ F( , ) (4.2-5) λf λf j (x2 + y 2 )(d − f ) x y g(x, y) = exp[− jk(d + f )]exp jπ F( , ) (4.2-8) λf λf 2 λf λf 2 1 x y I(x, y) = F( , ) (4.2-9) | λf |2 λf λf j x y If d = f , g(x, y) = exp[− j2kf ]F( , ) (4.2-10) λf λf λf Fourier transform using a lens is valid in Fresnel approximation (only radius at the output is limited). Without the lens, we need Fraunhofer approximation (radii at both output and input are limited). 4.3 Diffraction of Light Light not simply blocked by an opaque object, as in Ray Optics. It depends on the wavelength, the dimension of the object, and the distance between the object and the observation plane. A. Fraunhofer Diffraction x y Aperture function p(x, y) , with Fourier components P(ν ,ν ) = P( , ). x y λd λd 7 EE 485, Winter 2004, Lih Y. Lin Assume the incident wave is a plane wave of intensity Ii in z-direction. Using Eq. (4.2-1), Fraunhofer approximation, we obtain: I x y 2 I(x, y) = i P( , ) (4.3-4) (λd)2 λd λd → Proportional to the squared magnitude of the Fourier transform of the x y aperture function p(x, y) evaluated at the spatial frequency ν = , ν = . x λd y λd Example: Fraunhofer diffraction from a circular aperture 2 2 πD 2 2J (πDρ λd) I(ρ) = I 1 (4.3-7) 4λd i πDρ λd ρ = λ → Airy pattern. Center disk (Airy disk) has radius s 1.22 d / D , subtending an angle θ =1.22λ / D . B. Fresnel Diffraction At small distance (d → 0 ), the diffraction pattern is the shadow of the aperture. At medium distance (Fresnel diffraction), the diffraction pattern is the convolution of the aperture. Using Eq. (4.1-14), free-space propagation as a convolution, we obtain: 2 I (x − x')2 + (y − y')2 I(x, y) = i ∫∫ p(x', y')exp− jπ dx'dy' (4.3-11) (λd)2 λd At large d, the diffraction pattern becomes Fraunhofer diffraction pattern. The far field has an angular divergence proportional to λ / D , where D is the diameter of the aperture. 8 EE 485, Winter 2004, Lih Y. Lin 4.4 Image Formation Spatial filtering Two-lens imaging system (4-f system).