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. 

Casio fx-CG 50: Fresnel's Equations - Polarized Light

Casio fx-CG 50:  Fresnel's Equations - Polarized Light

Introduction:  Reflection and Transmission

The object of Fresnel's Equations is to calculate the reflection and transmissions of a light wave when it hits a change of medium.   Where the change of medium occurs is known as of a plane of incidence.  The equations calculate components:  p-polarized light for light parallel to the plane of incidence, and s-polarized light for perpendicular to the plane of incidence.  

The program presented also calculates Brewster's Angle, the angle required for light to hit the plane of incidence to have no reflection.  

Simplified, the equations are:

Angle of Transmission per Snell's Law:

Θt = arcsin(n1 × sin Θi ÷ n2)

Reflection Coefficients:

r ∥ = tan(Θi - Θt) ÷ tan(Θi + Θt) 

r ⟂ = -sin(Θi - Θt) ÷ sin(Θi + Θt) 

Transmission Coefficients:

t ∥ = (2 × sin Θt × cos Θi) ÷ (sin(Θi + Θt) × cos(Θi - Θt))

t ⟂ = (2 × sin Θt × cos Θi) ÷ sin(Θi + Θt)

Brewster's Angle:

Θb = arctan(Θt ÷ Θi)

Casio fx-CG 50 Program:  POLARIZE

(284 bytes)

Text file:

'ProgramMode:RUN

"2022-10-19 EWS"

Deg

"REFRACT INDEX 1"?->M

"_Theta_I"?->P

"REFRACT INDEX 2"?->N

"_Theta_T"

sin^-1 (M*sin P/N)->QDisps

"REFLECTION:"

"R_#E6D7_:"

tan (P-Q)/tan (P+Q)->RDisps

"R_#E6D5_:"

(-)sin (P-Q)/sin (P+Q)->TDisps

"TRANSMISSION:"

"T_#E6D7_:"

(2*sin Q*cos P)/(sin (P+Q)*cos (P-Q))->SDisps

"T_#E6D5_:"

(2*sin Q*cos P)/sin (P+Q)->UDisps

"BREWSTERS ANGLE:"

tan^-1 (N/M)->B

Program Code:

"2022-10-19 EWS"

Deg

"REFRACT INDEX 1"?→M

"ΘI"?→P

"REFRACT INDEX 2"?→N

"ΘT"

sin^-1 (M*sin P/N)→Q ◢

"REFLECTION:"

"R∥:"

tan (P-Q)/tan (P+Q)→R ◢

"R⟂:"

-sin (P-Q)/sin (P+Q)→T ◢

"TRANSMISSION:"

"T∥:"

(2*sin Q*cos P)/(sin (P+Q)*cos (P-Q))→S ◢

"T⟂:"

(2*sin Q*cos P)/sin (P+Q)→U ◢

"BREWSTERS ANGLE:"

tan^-1 (N/M)→B

The symbols ∥ and ⟂ are from the CHAR menu.  

Examples

All angles are in degrees.  

Example 1:

Inputs:

n1 = 1.00293 

ΘI = 20

n2 = 1.52 (glass)

Results:

ΘT = 13.04242914

R∥ = 0.1876099408

R⟂ = -0.2221588123

T∥ = 0.7836116039

T⟂ = 0.7778411877

Brewster's Angle = 56.58227879

Example 2:

Inputs:

n1 = 1.000293 (air)

ΘI = 34

n2 = 1.333 (water)

Results:

ΘT = 24.81075448

R∥ = 0.09793113211

R⟂ = -0.1866780705

T∥ = 0.8238955934

T⟂ = 0.8133219295

Brewster's Angle = 53.11516797

Sources

"Brewster's Angle".  Wikipedia.  https://en.wikipedia.org/wiki/Brewster%27s_angle  Retrieved October 18, 2022

"Fresnel's Equations: Reflection and Transmission"  HyperPhysics. http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/freseq.html  Retrieved October 16, 2022

Until next time,

Eddie

All original content copyright, © 2011-2022.  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.