foueier optics课件c3 foundations of scalar diffraction theory.ppt

CHAPTER 3 Foundations of Scalar Diffraction Theory,Review of physical optics Introduction of Fourier analysis in diffraction study,Real and complex expression of optical fields,For a monochromatic wave, the scalar field may be written explicitly as,A more compact form,Complex amplitude or phasor,Review of physical optics_1,Model of scalar diffraction,Review of physical optics_2,Unified form of scalar diffraction,Review of physical optics_3,The aperture is illuminated by an infinitely distant point source producing normally incident plane wave cos(n,r01)=cos,Review of physical optics_4,Comparison of three types of diffraction Kirchhoff solution is the arithmetic average of the two Rayleigh-sommerfeld solutions Kirchhoff solution and the two Rayleigh-Sommerfeld solutions to be essentially the same provided the aperture diameter is much greater than a wavelength,Generalization to nonmonochromatic waves,The more general case than monochromatic field is nonmonochromatic disturbance Attention is restricted to the predictions of the first Rayleigh-Sommerfeld solution, but similar results can be obtained for the other solutions,Temporal spectra of disturbances around an aperture,Scalar disturbance u(Po, t) observed behind an aperture in an opaque screen Disturbance u(P1, t) is incident on that aperture,Linear combination of monochromatic functions,This implies that nonmonochromatic time functions u(PI, t) and u(Po, t) are linear combination of monochromatic time functions,By change of variables = - , the above equations become,Nonmonochromatic u(PI, t) and u(Po, t),Recall first Rayleigh-Sommerfeld diffraction,For one time frequency the above equation can be written,Relation of u(PI, t) and u(Po, t),By identity,We get the relation,The disturbance at point Po is seen to be linearly proportional to the time derivative of the disturbance at each point P1 on the aperture,Monochromatic and nonmonochromatic waves,Diffraction of monochromatic waves can be used directly to synthesize the results for much more general nonmonochromatic waves. The monochromatic results are directly applicable themselves when the optical source has a sufficiently narrow spectrum,See diffraction from Fourier analysis,Diffraction can be formulated equally by Electronagnetic theory (Frenel-Kirchhoff, Rayleigh-Sommerfeld diffraction) and Fourier analysis The complex field distribution of a light wave can be analyzed into Fourier components across any plane The various spatial Fourier components can be identified as plane waves traveling in different directions away from that plane,Fourier analysis of light field,Across the z = 0 plane, the function U has a two-dimensional Fourier transform given by,Its inverse transform of its spectrum,Plane wave,A plane wave propagates with wave vector , where has direction cosines (,),Complex phasor amplitude of the plane wave,The coordinates of observation point are,Expand the wave vector,Complex phasor amplitude of the plane wave across a constant z-plane is,Simple form of complex amplitude,Expression by spatial frequency fx and fy,Angular spectrum of U(x,y,0),Angular spectrum of U(x,y,0),Decomposition of U(x,y,0),Angular spectrum of U(x,y,z),If the relation between A(/,/,z) and A(/,/,0) can be found, then the effects of wave propagation on the angular spectrum of the disturbance will be evident,Characteristics of U(x,y,z),U (x,y,z) can be written,In addition, U must satisfy the Helmholtz equation,Solution of Helmholtz equation,A must satisfy the differential equation,An elementary solution of this equation can be written in the form,Another way to get the solution,General form of a plane wave,When z=0, the above equation becomes,Obviously, P(x,y,z) has relation with P(x,y,0),The impact of value,When the direction cosines satisfy,The effect of propagation over distance z is simply a change of the relative phases of the various components of the angular spectrum,Evanescent waves,When (,) satisfy,Since is a positive real number, these wave components are rapidly attenuated by the propagation phenomenon,The meaning of evanescent wave,=cosx, = cosz x(z) fx=/ =frequency=details No conventional imaging system can resolve a periodic structure with a period that is finer than the wavelength of the radiation used,z,x,k,z,x,Amplitude transmittance function,An infinite opaque screen containing a diffracting structure is introduced in the plane z = 0 Transmitted field amplitude Ut(x, y;0) and incident field amplitude Ui(x, y; 0) at each (x, y) in the z = 0 plane Amplitude transmittance function of the aperture,Effects of a Diffracting Aperture on the Angular Spectrum,The relation of angular spectrums,If the diffracting structure is an aperture that limits the extent of the field distribution, the result is a broadening of the angular spectrum of the disturbance,Propagation of angular spectrum,The disturbance observed at (x, y, z) can be written in terms of the initial angular spectrum by inverse transforming,Consider propagation as a linear space-invariant system,Comparison of the above two equations shows that,Conclusion,Propagati