Sunday, December 4, 2022

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. 


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