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

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

3. …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

4. …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.

5. Con le condizioni iniziali:
Significa: …da cui…

7. Ricordiamo la Soluzione esatta:

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

10. Numerical solution

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

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

15. . . . 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.

16. Consider the Taylor Expansions:
Subtract…

17. Add…

18. 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.

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

20. Combine and solve for

21. Initial Condition

22. Propagation in free Space

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

24. 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.

25. Algebraically Solve for

26. Algebraically Solve for

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

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

29. l0 a L
ray optics
wave optics

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

31. 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…

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

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

34. Combine and solve for

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

36. 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.

37. 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.

38. 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)

39. 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:

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

41. Examples: single slit

42. Direction of
Propagation

43. Direction of
Propagation

44. Example2: double slit
Poisson Spot

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

47. Circular Aperture

48. Square Aperture

49. Square Aperture

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

51. 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: