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


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