I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Topic 2: Scalar Diffraction Aim: Covers Scalar Diffraction theory to derive Rayleigh-Sommerfled diffraction. Take approximations to get Kirchhoff and Fresnel approx- imations. Contents: 1. Preliminary Theory. 2. General propagation between two planes. 3. Kirchhoff Diffraction 4. Fresnel Diffraction 5. Summary O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -1- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Scalar Wave Theory Light is really a vector electomagnetic wave with E and B field linked by Maxwell’s Equations. Full solution only possible in limited cases, so we have to make as- sumptions and approximations. Assume: Light field can be approximated by a complex scalar potential. (am- plitude) Valid for: � λ , (most optical systems). Apertures and objects NOT Valid for: Very small apertures, Fibre Optics, Planar Wave Guides, Ignores Po- larisation. Also Assume: Scalar potential is a linear super-position of monochromatic compo- nents. So theory only valid for Linear Systems, (refractive index does not depend on wavelength). O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -2- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Scalar Potentials For light in free space the E and B field are linked by ∇ 0 : E = ∇ 0 : B = ∂ E ∇ ε 0 µ 0 ^ B = ∂ t � ∂ B ∇ ^ E = ∂ t which, for free space, results in the “Wave Equation” given by ∂ 2 E � 1 ∇ 2 E = 0 c 2 ∂ t 2 Assume light field represented by scalar potential Φ ( r ; t ) which MUST also obey the “Wave Equation”, so: ∂ 2 Φ � 1 ∇ 2 Φ = 0 c 2 ∂ t 2 Write the Component of scalar potential with angular frequency ω as ( r ) exp Φ ( r ; t ( ı ω t ) = u ) then substituting for Φ we get that, [ ∇ 2 + κ 2 = 0 ] u ( r ) where κ = 2 π = λ or wave number . ( r ) must obey Helmholtz Equation, (starting point for Scalar So that u Wave Theory). O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -3- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Diffraction Between Planes z y y x x z 0 u(x,y;z) u(x,y;0) P P 1 0 In P 0 the 2-D scalar potential is: ( x ; y ;0 ) = u 0 ( x ; y ) u where in plane P 1 the scalar potential is: ( x ; y ; z ) u where the planes are separated by distance z . ( x ; y ; z ) ( x ; y ) in plane P 0 we want to calculate in u Problem: Given u 0 is any plane P 1 separated from P 0 by z . Looking for a 2-D Scalar Solution to the full wave equation. O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -4- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Fourier Transform Approach Take the 2-D Fourier Transform in the plane, wrt x ; y . so in plane P 1 we get Z Z ; v ; z ) ( x ; y ; z ) exp ( � ı 2 π ( ux + vy )) d x d y ( u = U u since we have that Z Z ( x ; y ; z ) ; v ; z ) exp ( ı 2 π ( ux + vy )) d u d v = ( u u U ; v ; z ) in any plane, then we can easily find ( u then if we know U ( x ; y ; z ) the required amplitude distribution. u ( x ; y ; z ) is a Linear Combination of term of The amplitude u ; v ; z ) exp ( ı 2 π ( ux ( u + vy )) U These terms are Orthogonal (from Fourier Theory), So each of these terms must individually obey the Helmholtz Equa- tion. Note: the term: ; v ; z ) exp ( ı 2 π ( ux ( u + vy )) U ; v ; z ) and Direction = κ ; v = κ ) , ( u ( u in a Plane Wave with Amplitude U = 2 π = λ . where κ O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -5- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Propagation of a Plane Wave From Helmholtz Equation we have that [ ∇ 2 + κ 2 ; v ; z ) exp ( ı 2 π ( ux = 0 ] U ( u + vy )) Noting that the exp () terms cancel, then we get that � 1 ∂ 2 U � ; v ; z ) ( u + 4 π 2 � u 2 � v 2 ; v ; z ) = 0 ( u U ∂ z 2 λ 2 so letting = 1 γ 2 � u 2 � v 2 λ 2 then we get the relation that ∂ 2 U ) 2 U ( 2 πγ = 0 + 8 u ; v ∂ z 2 With the condition that ; v ;0 ( u ) = U 0 ( u ; v ) = F f u 0 ( x ; y ) g U we get the solution that ; v ; z ) ; v ) exp ( ı 2 πγ z ) ( u = U 0 ( u U This tells us how each component of the Fourier Transform propa- gates between plane P 0 and plane P 1 , so: Z Z ( x ; y ; z ) ; v ) exp ( ı 2 πγ z ) exp ( ı 2 π ( ux + vy )) d u d v = ( u u U 0 which is a general solution to the propagation problems valid for all z . O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -6- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Free Space Propagation Function Each Fourier Component (or Spatial Frequency) propagates as: ; v ; z ) ; v ) exp ( ı 2 πγ z ) ( u = U 0 ( u U Define: Free Space Propagation Function ; v ; z ) = exp ( ı πγ z ) H ( u so we can write: ; v ; z ) ; v ; z ) ; v ) H ( u = U 0 ( u ( u U ; v ; z ) . Look at the form of H ( u = 1 u 2 + v 2 + γ 2 λ 2 Case 1: If u 2 + v 2 = λ 2 then γ is REAL � 1 H ( u ; v ) Phase Shift so all spatial frequency components passed with Phase Shift. Case 2: If u 2 + v 2 = λ 2 them γ is IMAGINARY > 1 = exp ( � 2 π j γ H ( u ; v ) j z ) ; v ) with u 2 + v 2 = λ 2 decay with > 1 ( u So Fourier Components of U 0 Negative Exponential. (evanescent wave) O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -7- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Frequency Limit for Propagation � λ then the negative exponential decay will In plane P 1 where z remove high frequency components. u 2 + v 2 = λ 2 ; v ; z ) = 0 > 1 ( u U ( x ; y ; z ) is of limited extent. so Fourier Transform of u = 1 = λ , this corresponds to a grat- Maximum spatial frequency when u ing with period λ . d θ d sin θ = n λ > λ for d d f < λ . Information not transferred to plane P 1 . No diffraction when d O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -8- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Convolution Relation We have that ; v ; z ) ; v ; z ) U 0 ; v ; z ) ( u = H ( u ( u U so by the Convolution Theorem, we have that ( x ; y ; z ) ( x ; y ; z ) ( x ; y ; z ) = h � u 0 u ( x ; y ; z ) the distribution in P 1 due to u 0 ( x ; y ; z ) in P 0 . With u We then have that Z Z ( x ; y ; z ) exp ( ı 2 πγ z ) exp ( ı 2 π ( ux + vy )) d u d v = h < 1 = λ 2 u 2 + v 2 where r 1 � u 2 + v 2 γ = λ 2 “It-Can-Be-Shown” (with considerable difficulty), that � 2 π � exp ( ı κ r � ∂ ) ( x ; y ; z ) = h λ 2 ∂ z κ r where be have that = 2 π r 2 = x 2 + y 2 + z 2 κ and λ we therefore get that � 1 � exp ( ı κ r = 1 � � z ) ( x ; y ; z ) � ı h λ κ r r r which is known as “The Impulse Response Function for Free Space Propagation” O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -9- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Point Object y y x x r δ( ) a x,y z 0 P 0 u(x,y;z) P 1 In P 0 we have a Delta Function, so: = a δ ( x ; y ) ( x ; y ) u 0 So in plane P 1 we have ( x ; y ; z ) ( x ; y ; z ) = u ah � 1 � exp ( ı κ r � � z ) a = � ı λ κ r r r where r is the distance from: ( 0 ; 0;0 ( x ; y ; z ) ) ) So the intensity in plane P 1 is given by: � 1 = b 2 � � 2 � z ( x ; y ; z ) + 1 i r 2 κ 2 r 2 r where the λ 2 has been incorperated into the constant b . O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -10- Autumn Term C E P I S A Y R H T P M f E o N T
I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Shape of Function ( x ; 0; z ) is shown below: The shape of i 1.2 i(x,1) i(x,2) i(x,3) 1 i(x,4) 0.8 0.6 0.4 0.2 0 -4 -3 -2 -1 0 1 2 3 4 = λ ; 2 λ ; 3 λ ; 4 λ , x -scale in λ s. for z Compare with a isolated 3-D point source, spherical expanding waves, so intensity in plane at distance z of: = b 2 ( x ; y ; z ) I r 2 1 I(x,1) 0.9 I(x,2) I(x,3) 0.8 I(x,4) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -4 -3 -2 -1 0 1 2 3 4 Note We have a 2-D Delta Function, (whole in a screen), and NOT an 3-D point source. O P T I C D S E G I R L O P P U A P D S Scalar Diffraction -11- Autumn Term C E P I S A Y R H T P M f E o N T
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