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Michael Scalora U.S. Army Research, Development, and Engineering Center Redstone Arsenal, Alabama, & Universita' di Roma "La Sapienza" Dipartimento di Energetica OPTICS BY THE NUMBERS LOttica Attraverso i Numeri Rome, April-May 2004

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Soluzione Numeriche di Equazioni Nonlineari Accoppiate Usando il Predictor-Corrector Algorithm: More Examples

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…is just a Taylor expansion for ANY function …always finds a second order accurate solution to the generic differential equation

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Keep in mind that an n th order differential equation can always be reduced to n coupled equations of first order. Example 1: can be rewritten as the system: …with appropriate initial conditions.

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Con le condizioni iniziali: Significa: …da cui…

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Ricordiamo la Soluzione esatta:

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Oscillatori smorzati accoppiati Example 2: damping restoring force driving force

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Numerical solution

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Calculation of numerical derivatives However, in reality t cannot be zero, and so one incurs into an error: ?

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Consider the Taylor Expansion: Then: Calculating the numerical derivative as: Implies an error of order t …

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Usando la matematica del buon senso La derivata al punto puo essere definita in almeno due modi: mediando le due soluzioni…... …si presume con un errore piu piccolo.

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Consider the Taylor Expansions: Subtract…

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Add…

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1.Increasing precision requires more information, storage 2. An n th order differential equation can always be reduced to n coupled equations of first order. 3. Calculating derivatives and integrating differential equations is more of an art than a science, and one uses whatever works. Care should be exercised when considering functions that vary rapidly in space or time. Ex.: In general,

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We have all the elements for a simple one dimensional electromagnetic pulse propagation algorithm

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Combine and solve for

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Initial Condition

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Propagation in free Space

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n=1 n=2 Incident transmitted reflected |E| 2

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Another point of view: Lequazione di secondo grado: Diventa due equazioni accoppiate di primo grado… …che sono le equazioni di Maxwell in una dimensione e il tempo cioe il punto di partenza.

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Algebraically Solve for

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1. The SVEA, the Wave Equation, and Diffraction 2. Spectral Methods and Free-Space Diffraction

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Diffraction The Bending of Light Around Corners a a L L ray optics wave optics

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a a L ray optics wave optics

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E This term decribes diffraction Assumption: the beam envelpe does not vary in time so-called CW (continuous wave) beam

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For simplicity, lets assume only one transverse dimension: Assume… E(x,z) e un inviluppo che varia lentamente nello spazio rispetto a e nel tempo rispetto a /c. Apply the SVEA…

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We may also define the Fresnel number as… Il fascio non diffrange: Ray optics.. Regime di Forte diffrazzione.

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How to calculate diffraction using the wave equation Confrontiamo… Con…

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Combine and solve for

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We can proceed as follows: Substitute and solve for Let…

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Ci sono altri modi di procedere che non richiedono unimpostazione cosi onerosa dal punto di vista del numero di varibili e vettori di cui tener conto. Ci occuperemo di metodi cosidetti spettrali.

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How to calculate diffraction using the wave equation… Useful properties of the Fourier Transform: Derivatives… N is the number of points used to discretize the space x in units x.

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Some Advantages of using Spectral (FT) methods (i)calculation of all kinds of derivatives is simple (ii)derivatives are extremely accurate (iii) simple algorithm Some Disadvantages (i)It is slower (ii)Functions should be very smooth (which usually are)

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…is the solution in q-space. To find the solution in x-space, where things are observable, we need to take the inverse transform: Taking the FT of both sides with respect to x… And…

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Algorithm: (i)Trasformata di Fourier (FT) del campo iniziale (ii)Moltiplicazione per il propagatore (iii)Transformata inversa di tutto

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Examples: single slit

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Direction of Propagation

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Direction of Propagation

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Example2: double slit Poisson Spot

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Per aperture con geometrie piu complicate, e.g. Apertura Circolare o quadrata, e necessario Ritornare alle due dimensioni trasverse:

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Circular Aperture

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Square Aperture

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Algorithm: (i)Trasformata di Fourier (FT) del campo iniziale (ii)Moltiplicazione per il propagatore (iii)Transformata inversa di tutto

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Removing the second order spatial derivative also means making the Paraxial Wave Approximation: the beam radius cannot be too small or inconsistencies with experimental observations may arisem since the wave tends to diffract very fast, contrary to expectations. This problem may be partially cured as follows:

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