OPTICS BY THE NUMBERS L’Ottica Attraverso i Numeri Michael Scalora U.S. Army Research, Development, and Engineering Center Redstone Arsenal, Alabama, 35898-5000 & Universita' di Roma "La Sapienza" Dipartimento di Energetica Rome, April-May 2004

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

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

…with appropriate initial conditions. Keep in mind that an nth 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.

Con le condizioni iniziali: Significa: …da cui…

Ricordiamo la Soluzione esatta:

Oscillatori smorzati accoppiati Example 2: driving force restoring force damping Oscillatori smorzati accoppiati

Numerical solution

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

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

. . . 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.

Consider the Taylor Expansions: Subtract…

Add…

In general, Increasing precision requires more information, storage 2. An nth order differential equation can always be reduced to n coupled equations of first order. Ex.: 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.

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

Combine and solve for

Initial Condition

Propagation in free Space

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

L’equazione di secondo grado: Another point of view: L’equazione 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.

Algebraically Solve for

Algebraically Solve for

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

Diffraction The Bending of Light Around Corners l0 l0 a a L ray optics wave optics L

l0 a L ray optics wave optics

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

E(x,z) e’ un inviluppo che varia lentamente For simplicity, let’s assume only one transverse dimension: Assume… E(x,z) e’ un inviluppo che varia lentamente nello spazio rispetto a l, e nel tempo rispetto a t=l/c. Apply the SVEA…

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

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

Combine and solve for

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

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

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 dx.

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

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

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

Examples: single slit

Direction of Propagation

Direction of Propagation

Example2: double slit Poisson Spot

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

Circular Aperture

Square Aperture

Square Aperture

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

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: