Saturday, December 2, 2023

Swiss Micros DM41X and Numworks: Fresnel Reflection Calculations

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


All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


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