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An Overview on the Huygens-Fresnel Principle, Coherence and van CitterZernike Theorem References: 1.Modeling and Simulation of Beam Control Systems: Part 1. Foundations of Wave Optics Simulation www.mza.com/publications/MZADEPSBCSMSCP3.ppt 2.Professor David Attwood (Univ. California at Berkeley), AST 210/EECS 213, Lecture 16, http://ast.coe.berkeley.edu//sxreuv/2005/Ch08C.pdf 1. Huygens-Fresnel Principle of Wave Propagation Fundamental theory of wave propagation • • • Wave equation (monochromatic) in vacuum or uniform dielectric medium (1) Wave equation in presence of fluctuations n(x,y,z; t): third term couples the polarizations during propagation Fundamental approximation: order of magnitude calculations imply that the coupling term is negligible. In this approx., the fluctuations do not mix polarization components Turbulent prop still satisfies the “scalar diffraction” picture. Resulting equation, with extra decomposition n(r) = <n>+dn(r), and letting k = k0 n0 = averaged wave vector in unperturbed medium (2) (3) perturbation term relative to Eq (1) Scalar Diffraction Theory When monochromatic light propagates through vacuum or ideal dielectric media, the spatial and temporal variations of the electromagnetic field can be separated, and the spatial variations of the six components of the electric and magnetic field vectors are identical. The spatial variation of the two vector fields, E and B, can therefore be represented in terms of a single scalar field, u. electromagnetic field scalar field Non-monochromatic light can be expressed as a superposition of monochromatic components: The Huygens-Fresnel Principle The propagation of optical fields is described by the Huygens-Fresnel principle, which can be stated as follows: Knowing the optical field over any given plane in vacuum or an ideal dielectric medium, the field at any other plane can be expressed as a superposition of “secondary” spherical waves, known as Huygens wavelets, originating from each point in the first plane. u2 u1 R r1 r2 q z1 z2 Huygens wavelets The Fresnel Approximation When the transverse extents of the optical field to be propagated are small compared with the propagation distance, we can make small angle approximations, yielding useful simplifications. u2 u1 r1 R r2 q z1 z2 The Fresnel Approximation Conditions for Validity The Fresnel approximation is based upon the assumption |r2-r1| << Dz. Here r1 and r2 represent the transverse coordinates in the initial and final planes for any pair of points to be considered in the calculation. What pairs of points must be considered depends upon the specific problem to be modeled. This requirement will be satisfied if the transverse extents of the region of interests at the two planes are sufficiently small, as compared to the propagation distance. The requirement can also be satisfied if the light is sufficiently wellcollimated, regardless of the propagation distance. The Fresnel approximation can also be used, in a modified form, for light that is known to approximate a known spherical wave, such as the light propagating between the primary and secondary mirrors of a telescope. Fourier Optics When the Fresnel approximation holds, the Fresnel diffraction integral can be decomposed into a sequence of three successive operations: 1. Multiplication by a quadratic phase factor 2. A scaled Fourier transform 3. Multiplication by a quadratic phase factor. quadratic phase factor scaled Fourier transform quadratic phase factor The Fourier Transform Physical Interpretation of the Fresnel Diffraction Integral u2 u1 r2=0 r1=0 z1 z2 Equivalently, the quadratic phaseappearing factors can The two quadratic phase factors in be the thought diffraction of as two Huygens wavelets, to originating Fresnel integral correspond two from the points (r1=0, z=z1) and (r2=0, z=z2). confocal surfaces. Fourier Optics in Operator Notation For notational convenience it is sometimes useful to express Fourier optics relationships in terms of linear operators. We will use PDz to indicate propagation, FDz for a scaled Fourier transform, and QDz for multiplication by a quadratic phase factor. Multi-Step Fourier Propagation z1 It is sometimes useful to carry out a Fourier propagation in two or more steps. Dz z2 The individual propagation steps may be of any size and in either direction. z1 z2 Fourier Optics: Some Examples (all propagations between confocal planes) circ rect(x)rect(y) Gaussian FDz FDz FDz Airy pattern sinc(x)sinc(y) Gaussian Waves vs. Rays Scalar diffraction theory and Fourier optics are usually described in terms of waves, but they can also be described, with equal rigor, in terms of rays. This may seem surprising, because rays are constructs more typically associated with geometric optics, as opposed to wave optics. In geometric optics, rays are thought of as carrying an energy, possibly distributed over a range of wavelengths. In wave optics, each ray must be thought of as carrying a certain complex amplitude, at a specific wavelength. The advantage of thinking in terms of rays, as opposed to waves, is that it makes it easier to take into account geometric considerations, such as limiting apertures. A wave can be thought of as a set of rays, and geometric considerations may allow us to restrict our attention to a smaller subset of that set. A Wave as a Set of Rays u2 u1 r2 r1 z1 z2 Suppose we collect all impinging onlight the point zcan from all Repeating Each From ray Each Huygen’s the defines Huygens-Fresnel thenow procedure wavelet the contribution can forthe principle, be all rays points further from any in adecomposed point the (scalar) any source light into wave at wave ra1 set to the ofbe rays, be field 2can points decomposed decomposed connecting in thethe first into into origination plane. ar a2set set of This spherical spherical point setrof rays waves waves theisplane (Huygen’s equivalent (Huygen’s z withwavelets) to wavelets) alla points Huygen’s on at a specific point onof the plane 1zon 2. Conversely,1 the same ray also originating originating some other from from plane all alloriginating zthe the points on one one plane, plane, zat z11..rgoing wavelet, this time the point r2 and backwards. defines the contribution from at aon point source field at r1. 2. points 2 to the Waves vs. Rays Mathematical Equivalence Ray picture: that the Note that the field at each point r2 isRecall expressed as “wave the picture” equations were superposition of the contributions from all points r1. derived from the “ray picture” equation with no additional assumptions. Wave picture: Note that the field u2 at all points is expressed in terms of the field u1 at all points. Waves vs. Rays Why the “Ray Picture” is Useful Thinking of light as being made up of rays, as opposed to waves, makes it easier to take into account a priori geometric constraints pertaining to two or more planes at the same time. For example, if the light to be modeled is known to pass through a limiting apertures, we can restrict our attention to just the set of the rays that pass through that aperture. Similarly, if there are multiple limiting apertures, we can restrict our attention to the intersection of the ray sets defined by the individual apertures. It is important to understand that strictly speaking a given ray set remains well-defined only within a contiguous volume filled with a uniform dielectric medium, and only for purely monochromatic light. Extending Scalar Diffraction Theory Relatively easy / cheap Monochromatic Coherent Uniform polarization Ideal media a a a a Quasi-monochromatic Temporal partial coherence Non-uniform polarization Phase screens, gain screens Harder / more expensive • • • • Broadband illumination Spatial partial coherence Ultrashort pulses Wide field incoherent imaging Scalar Diffraction Theory and Fourier Optics Scalar Diffraction Theory: the electric and magnetic vector fields are replaced by a single complex-valued scalar field u. The Huygens-Fresnel Principle: knowing the field at any plane, the field at any other plane can be expressed as a superposition of spherical waves originating from each point in the first plane. The Fresnel Approximation: for |r|<<|Dz|, the equations simplify. Fourier Optics: the propagation integral can be expressed in terms of Fourier transforms and quadratic phase factors. Waves vs. Rays: light waves can be thought of as sets of rays, where each ray carries a complex amplitude. Extending Fourier Optics: it is possible. The Discrete Fourier Transform What happens when we try to represent a continuous complex field on a finite discrete mesh? How can we reconstruct the continuous field from the discrete mesh? How can we ensure that the results obtained will be correct? What can go wrong? Reference: The Fast Fourier Transform, by Oran Brigham The DFT as a Special Case of the Fourier Transform F window F sample F repeat F DFT pair The DFT as a Special Case of the Fourier Transform u u rect repeat (u ×rect) F F F Constructing the Continuous Analog of a DFT Pair uD One When Now way that using to wedo DFTs, have this obtained in doorder this isto a If one applies Fourier minimize to new use DFT Fourier the pair,computational interpolation: we can iterate. interpolation to bothoften a function requirements, one To With interpolate each iteration, the function, thethe mesh and its DFT transform, chooses to make the mesh zero-pad spacingsinterpolated its in each transform, domain then resulting versions spacing as large as possible compute decrease, the and the pair. mesh DFT. do notstill form ainverse DFT while obtaining correct extents increase, all by the results. (Nyquist However if wewhile then perform To interpolate theCriterion) transform, same factor, the mesha second Fourier interpolation zero-pad the Sometimes dimension (N) it function, isincreases useful then to by in each domain and average compute the DFT. construct the square a of more thatdensely factor. the results from the two-steps, the sampled version of the result is aand/or DFT pair. function its transform. FD (uD) uD’ FD(uD’) 1 2 1 2 1 2 1 2 New DFT Pair Constructing the Continuous Analog of a DFT Pair Example: A Discrete “Point Source” N=16 u F(u) N=64 N=256 The Nyquist Criterion The Whitaker-Shannon Sampling Theorem shows that it is possible to exactly recover a continuous function from a discretely sampled version of that function if and only if (a) the function is strictly band-limited and (b) the sample spacing satisfies the Nyquist Criterion: the spacing must be less than or equal to half the period of the highest frequency component present. In the context of wave optics simulation the Nyquist criterion defines the maximum mesh spacing that will suffice to represent a given optical field: Here qmax is the band-limit of the complex field to be represented on the discrete mesh when we compute the DFT in the course of performing a DFT propagation. Note that this step occurs only after we have multiplied the field by a quadratic phase factor: The Nyquist Criterion Wave Optics Example u1 θmax θmax D1 2θmax 1 u2 2 1 2θmax D2 2 d1 d2 z1 z2 Aliasing If we attempt to represent a field with energy propagating at angles exceeding the Nyquist limit for the given mesh spacing, that energy will instead show up at angles below the Nyquist limit; this phenomenon is called aliasing. The Discrete Fourier Transform What happens when we try to represent a continuous complex field on a finite discrete mesh? We lose any energy falling outside the mesh extents in either domain. Discrete sampling in one domain implies periodicity in the other. How can we reconstruct the continuous field from the discrete mesh? DFT interpolation. (Or, to obtain a new DFT pair, a somewhat more complicate procedure involving two DFT interpolations.) How can we ensure that the results obtained will be correct? By enforcing the Nyquist criterion. What can go wrong? Aliasing 2. Optical Coherence and van CitterZernike Theorem principle q tan 1 (r z )