Appendix J: N-Slit Interferometric Calculations—Numerical Approach
J.1 Introduction
The first numerical calculations representing the interferometric equation
(J.1) |
and its associated geometry was performed using Fortran IV (Duarte and Paine 1989; Duarte 1991, 1993). Then the program was transitioned to Visual Fortran (Duarte, 2002). MATLAB® versions of the calculations have been developed more recently Duarte et al. (2013). Here, we provide a MATLAB version of a simple program that deals with the experimental situation related to the basic probability amplitude
(J.2) |
In Duarte et al. (2013), the numerical calculations were performed using a MATLAB version of the program that allows cascade calculations from one interferometric plane to another interferometric plane as previously demonstrated in Duarte (1993), while using Fortran IV.
J.2 Program
This is a sample program, written in MATLAB language. It is a simplified version of the type of program used by Duarte et al. (2013). This program, applicable to a uniform array of N slits, assumes a slit separation equal to the slit width w and exact dimensions for the slit width and slit separation. The program uses the following input data:
1. Number of slits N
2. Slit width w in micrometers (μm)
3. Intra-interferometric distance D〈x|j〉 in meters (m)
4. Wavelength λ in nanometers (nm)
01%NSLITSGENA.m
02 clear all
03 datestr(now)
04 N = input(‘Number of slits N = ’);
05 w = input(‘Slit width in micrometers w = ’);
06 d = input(‘Slit to screen distance in meters d = ’);
07 lambda = input(‘Wavelength in nanometers lambda = ’);
08 d = d*1e6;
09 lambda = lambda/1e3;
10 NP = 40030;% Screen size in micrometers.
11 s = 2*pi/lambda;%wavenumber
12 j0 = 20015-w*(N-0.5);% left side left aperture position
13 j01 = j0+1;
14 j0n = j0+w;
15 for ii = 1:NP, % Screen
16 for j = j01:j0n,%Field due to left slit on screen.
17 r = sqrt((ii-j)∧2+d∧2);
18 E(j) = 1e3*(i*d/lambda)*(exp(i*s*r))/r∧2;
19 end
20 ii10 = ii/1000-round(ii/1000);
21 if ii10 = =0
22 iactual = [ii NP]
23 else
24 end
25 EF(ii) = sum(E);
26 E12(ii) = sum(E)*conj(sum(E));
27 E1(ii) = abs(sum(E));%(module)
28 E2(ii) = angle(sum(E));%(angle)
29 end
30 for k = 1:N
31 for ii = 1:NP
32 EF1(k,ii+2*w*(k-1)) = EF(ii);
33 FI2(k,ii+2*w*(k-1)) = E12(ii);
34 FIR(k,ii+2*w*(k-1)) = E1(ii);
35 fir(k,ii+2*w*(k-1)) = E2(ii);
36 end
37 end
38 HQ = sum(FI2);
39 IIMAX = NP+(N-1)*2*w;
40 H = (1:IIMAX)*0;
41 for j = 1:N
42 for m = (j+1):N,
43 H = H+2*FIR(j,:).*FIR(m,:).*cos(fir(m,:)-fir(j,:));
44 end
45 end
46 for j = 1:N
47 FIRx(j,:) = FIR(j,:).*cos(fir(j,:));%REAL PART
48 FIRy(j,:) = FIR(j,:).*sin(fir(j,:));%IMAGINARY PART
49 end
50 FIREX = sum(FIRx);
51 FIREY = sum(FIRy);
52 Efield1(1,:) = FIREX;
53 Efield1(2,:) = FIREY;
54 save Efield1 Efield1 -ascii
55 ESQ = H+HQ;
56 plot(ESQ,‘r’)
57 hold on
58 datestr(now)
The specifics of this program differ from the descriptive approach disclosed in Appendix I. Here, ii is the position on the interferometric plane (screen) and j the position at the slit array. Also, on line 18, and associated definitions, a particular wave function amplitude is assumed. It should be noted that the formalism, defined in Equation J.1, tolerates various forms of wave function amplitudes. In this regard, the search for the simplest and most suitable wave function amplitude, as determined by comparisons with measured interferograms, is ongoing.
References
Duarte, F. J. (1991). Dispersive dye lasers. In High Power Dye Lasers (Duarte, F. J., ed.). Springer, Berlin, Germany, Chapter 2.
Duarte, F. J. (1993). On a generalized interference equation and interferometric measurements. Opt. Commun. 103, 8–14.
Duarte, F. J. (2002). Secure interferometric communications in free space. Opt. Commun. 205, 313–319.
Duarte, F. J. and Paine, D. J. (1989). Quantum mechanical description of N-slit interference phenomena, in Proceedings of the International Conference on Lasers’88, (Sze, R. C. and Duarte, F. J., eds.). STS Press, McLean, VA, pp. 42–27.
Duarte, F. J., Taylor, T. S., Black, A. M., and Olivares, I. E. (2013). Diffractive patterns superimposed over propagating N-slit interferograms. J. Mod. Opt. 60, 136–140.