Swiss Micros DM41X and Numworks: Fresnel Reflection Calculations
Introduction
Given the following:
ni: incident refractive index (input)
nt: incident transmitted index (output)
θi: angle of incidence, in degrees
The following are calculated:
θt: angle of transmission, in degrees
θt = arcsin(ni × sin θi ÷ nt)
θb: angle as determined by Brewster's law
θb = arctan(nt ÷ ni)
R: power reflectance coefficient
R = (ni - nt)^2 ÷ (ni + nt)^2
T: power transmitted coefficient
T = 1 - R
Swiss Micros DM41X Code: FRES
01 LBL^T FRES
02 DEG
03 ^T N.I?
04 PROMPT
05 STO 01
06 CLA
07 29
08 XTOA
09 ^T |- .I?
10 PROMPT
11 STO 03
12 ^N.T?
13 PROMPT
14 STO 02
15 RCL 01
16 RCL 02
17 /
18 RCL 03
19 SIN
20 *
21 ASIN
22 STO 04
23 CLA
24 29
25 XTOA
26 ^T |- .T=
27 ARCL 04
28 AVIEW
29 STOP
30 RCL 02
31 RCL 01
32 /
33 ATAN
34 STO 05
35 CLA
36 29
37 XTOA
38 ^T |- BRW=
39 ARCL 05
40 AVIEW
41 STOP
42 RCL 01
43 RCL 02
44 -
45 RCL 01
46 RCL 02
47 +
48 /
49 X↑2
50 STO 06
51 ^T R=
52 ARCL X
53 AVIEW
54 STOP
55 1
56 X<>Y
57 -
58 STO 07
59 ^T T=
60 ARCL X
61 AVIEW
62 END
Numworks - Python Code: fres.py
from math import *
print("Fresnel Equations")
print("Electrical Fields")
print("(angles are in degrees)")
ti=float(input("\u03b8.i? "))
ai=radians(ti)
ni=float(input("n.i? "))
nt=float(input("n.t? "))
at=asin(ni/nt*sin(ai))
tt=degrees(at)
ab=atan(nt/ni)
tb=degrees(ab)
r=((ni-nt)/(ni+nt))**2
t=1-r
print("\n\u03b8.t =",tt)
print("Brewster's Law=\n",tb)
print("Reflect Coef.=\n",r)
print("Trans. Coef.=\n",t)
Example
Inputs:
θi: 46°
ni: 1.33
nt: 1.5
Outputs:
θt ≈ 39.6291°
θb ≈ 48.4377°
R ≈ 0.0036
T ≈ 0.9964
Sources
R. Paschotta, article on 'Fresnel equations' in the RP Photonics Encyclopedia, accessed on 2023-11-04. URL: https://www.rp-photonics.com/encyclopedia_cite.html?article=Fresnel%20equations
Woan, Gaham. The Cambridge Handbook of Physics Formulas 2003 Edition. Cambridge University Press. 2000. ISBN 978-0-511-07589-6
Eddie
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