FRESNEL'S EQUATIONS

Exploring Reflection and Refraction

Anthony F. Behof
DePaul University
Department of Physics
2219 N. Kenmore Avenue
Chicago, Illinois 60614
abehof@.depaul.edu

Introduction

Fresnel's equations describe the behavior of orthogonally polarized components of an electric field at an interface.  Since any electric field can be described by a linear combination of two orthogonally polarized components, the Fresnel equations are general expressions for the behavior of electric fields at an interface of two linear, isotropic, homogeneous media. To apply these expressions, we resolve the E-field into components parallel to, and perpendicular to, the plane of incidence.  The results of the interaction of the E-field with the interface are given in terms of the ratio of the amplitudes of the reflected and transmitted fields to the amplitude of the incident field.  In most cases, the media in question are linear dielectrics with magnetic permeability close to that of free space.  For these cases, the Fresnel equations can be written in this form:

Form 1:

\(\(\(r\_perp = \(\(\(n\_i\) cosθ\_i - \(n\_t\) cosθ\_t\)\/\(\(n\_i\) cosθ\_i + \(n\_t\) cosθ\_t\)\  = \(cosθ\_i - \(n\_ti\) cosθ\_t\)\/\(cosθ\_i + \(n\_ti\) cosθ\_t\)\)\)\(
\)
    \)\)
\(\(\(t\_perp = \(\(2 \( n\_i\) cosθ\_i\)\/\(\(n\_i\) cosθ\_i + \(n\_t\) cosθ\_t\)\  = \(2  cosθ\_i\)\/\(cosθ\_i + \(n\_ti\) cosθ\_t\)\)\)\(
\)
    \)\)
\(\(\(r\_para = \(\(\(n\_t\) cosθ\_i - \(n\_i\) cosθ\_t\)\/\(\(n\_i\) cosθ\_t + \(n\_t\) cosθ\_i\)\  = \(\(n\_t\) cosθ\_i - cosθ\_t\)\/\(cosθ\_t + \(n\_ti\) cosθ\_i\)\)\)\(
\)
    \)\)
\(t\_para = \(\(2 \( n\_i\) cosθ\_i\)\/\(\(n\_i\) cosθ\_t + \(n\_t\) cosθ\_i\)\  = \(2  cosθ\_i\)\/\(cosθ\_t + \(n\_ti\) cosθ\_i\)\)\)

where (Subscript[r, perp]) and (Subscript[t, perp]) are the reflection and transmission coefficients of the  component of the incident field polarized perpendicular to the plane of incidence.  Similarly, (Subscript[r, para]) and (Subscript[t, para]) are the reflection and transmission coefficients of the parallel component.  The angles (Subscript[θ, i]) and (Subscript[θ, t]) are the angles of incidence and transmission, respectively. These are, of course, with repect to the normal of the interface.  The angle (Subscript[θ, r]) is not included explicitly because it is equal to (Subscript[θ, i]).  Finally, the value (Subscript[n, ti]) is the ratio of the index of refraction of the transmitting medium to that of the incident medium.
	Sometimes it is advantageous to eliminate (Subscript[n, ti]) from the Fresnel equations and have them in terms of (Subscript[θ, i]) and (Subscript[θ, t]) only.  Using Snell's law and some algebra, we can write Fresnel's equations in this Form 2.

Form 2:

\(\(\(r\_perp = \(-\(\(sin \((θ\_i - θ\_t)\)\)\/\(sin \((θ\_i + θ\_t)\)\)\)\)\)\(
\)
    \)\)
\(\(\(t\_perp = \(+\(\(2 \( sinθ\_t\) 
              cosθ\_i\)\/\(sin \((θ\_i + θ\_t)\)\)\)\)\)\(
\)
    \)\)
\(\(\(r\_para = \(+\(\(tan \((θ\_i - θ\_t)\)\)\/\(tan \((θ\_i + θ\_t)\)\)\)\)\)\(
\)
    \)\)
(Subscript[t, para]) = +((2  sin(Subscript[θ, t]) 
              cos(Subscript[θ, i]))/(sin ((Subscript[θ, i]) + (Subscript[θ, t])) 
              cos ((Subscript[θ, i]) - (Subscript[θ, t]))))

On the other hand, if we know the indices of refraction of both media and would prefer to eliminate the transmission angle (Subscript[θ, t]) from the equations, we can write Fresnel's equations in yet another form.

Form 3:

\(\(\(r\_perp = \(Cosθ\_i - \@\(n\_ti\%2 - Sinθ\_i\^2\)\)\/\(Cosθ\_i + \@\(n\_ti\%2 - Sinθ\_i\^2\)\)\)\(
\)
    \)\)
\(\(\(t\_perp = \(2\ Cosθ\_i\)\/\(Cosθ\_i + \@\(n\_ti\%2 - Sinθ\_i\^2\)\)\)\(
\)
    \)\)
\(\(\(r\_para = \(n\_ti\%2\ Cosθ\_i - \@\(n\_ti\%2 - Sinθ\_i\^2\)\)\/\(n\_ti\%2\ Cosθ\_i + \@\(n\_ti\%2 - Sinθ\_i\^2\)\)\)\(
\)
    \)\)
\(t\_para = \(2\ n\_ti\ Cosθ\_i\)\/\(n\_ti\%2\ Cosθ\_i + \@\(n\_ti\%2 - Sinθ\_i\^2\)\)\)

How To Interact With This Notebook

This Mathematica notebook is composed of different types of cells, some of which are interactive.  Cells such as this one are 'dead' -- they contain text only and no input.  Cells which contain Mathematica code are interactive.  The reader will see a text cell just  above the code cell with instructions to activate the cell and a description of what the result will be.  The reader activates a code cell by highlighting the blue cell bracket on the right side of the screen and pressing the "Insert" key This is the "Enter" key on the numeric keypad.  (Note: Highlighted cells may also be activated by pressing Shift-Return).

Example 1:

Activate the code cell below to produce a plot of the sine of x versus x, as x goes from zero to 2 Pi.

Plot[Sin[x], {x, 0, 2 π}]

You should now see the plot described above. If not, please re-read the instructions and try again. You will notice that Mathematica added an In and an Out label with some numbers after them. This is Mathematica's way (and ours) of knowing the sequence of operations that we performed during a session. For now, you do nothing with these. As an experienced user or programmer these labels are useful for troubleshooting a program.
	There is one other way in which the reader interacts with this notebook.  In order for Fresnel's equations to take on hard values, a value must be specified for the indices of refraction of the incident and transmitting media.  In this notebook, we will assume that one of the media is air (for which n = 1), so that we only need to specify a value for the index of the second medium.  It is left to the reader to choose this value.  Some commonly used values are n = 1.5 (certain types of glass) and n = 1.33 (water).  However, any value greater than or equal to 1 is physically plausible. The other interpretation is that n will be (Subscript[n, ti]) 
	To input the value you wish, activate the code cell below.  This will bring up a window into which you may type your value and click on  OK. Please be warned, though, that this specifies the value of n for the non-air medium for the entire notebook.  To change n, you must return to this cell and execute it again and then enter a new value.

Clear[n];
n = Input["Enter your value for n"]

Amplitude Coefficients

External Reflection

In this section, we will examine how the amplitude coefficients (Subscript[r, perp]), (Subscript[t, perp]), (Subscript[r, para]), and (Subscript[t, para]) change with respect to the angle of incidence.  Let us first take the case of fields incident in air at a dielectric interface.  Activate the following code cell to see a plot of the two reflection coefficients and two transmission coefficients versus (Subscript[θ, i]):

nti = n; 
rPerp[
        theta_] := ((Cos[theta] - (Sqrt[(nti^2) - Sin[theta]^2]))/(Cos[theta] + (Sqrt[(nti^2) - Sin[theta]^2]))); 
tPerp[
        theta_] := ((2 Cos[theta])/(Cos[theta] + (Sqrt[(nti^2) - Sin[theta]^2]))); 
rPara[
        theta_] := (((nti^2) Cos[theta] - (Sqrt[(nti^2) - Sin[theta]^2]))/(Cos[theta] (nti^2) + (Sqrt[(nti^2) - Sin[theta]^2]))); 
tPara[
        theta_] := ((2 nti Cos[theta])/(Cos[theta] (nti^2) + (Sqrt[(nti^2) - Sin[theta]^2]))); 
plot1 = 
      Plot[{rPerp[theta], rPara[theta], tPerp[theta], tPara[theta]}, {theta, 
          0, (π/2)}, Frame -> True, 
        FrameTicks -> {{0, (π/4), (π/2)}, {-1, -0.5, 0, 
              0.5, 1}}, PlotRange -> {{0, (π/2)}, {-1, 1}}, 
        DisplayFunction -> Identity]; 
\(Show[plot1, 
      Graphics[Text["(Subscript[t, para])", {π\/4, 0.9}]], 
      Graphics[Text["(Subscript[t, perp])", {π\/4, 0.55}]], 
      Graphics[Text["(Subscript[r, para])", {π\/8, 0.4}]], 
      Graphics[Text["(Subscript[r, perp])", {π\/8, \(-0.4\)}]], 
      Graphics[Text["(Subscript[θ, p])", {ArcTan[nti], \(-0.2\)}]], 
      DisplayFunction -> $DisplayFunction]\)

Note that, when the incident angle is close to zero (i.e., at near normal incidence), the two transmission coefficients are nearly equal, while the two reflection coefficients are nearly equal in magnitude but opposite in sign.  This is easily shown in the mathematics by setting (Subscript[θ, i]) = 0 in any form of Fresnel's equations.  The significance of this sign difference will be discussed in the next section.
	The other noteworthy feature of this plot is the fact that, at a certain value of (Subscript[θ, i]), (Subscript[r, para]) is zero.  This angle is known as Brewster's angle or the polarization angle.  The derivation of this second term should be clear:  At this angle, the parallel component of the reflected light is extinguished, and the reflected light is polarized in the direction perpendicular to the plane of incidence.

Internal Reflection

Now let's look at the case where the light is incident in the dielectric medium.  Assuming a piece of dielectric surrounded by air, the value (Subscript[n, ti]) will now be the reciprocal of the index of refraction of the dielectric.  Activate the following code cell to see a plot of the two reflection coefficients for this case.

nti'=(1/n);
q[theta_]:=If[(((nti'))^2)-Sin[theta]^2>0,((Cos[theta]-(Sqrt[(((nti'))^2)-Sin[theta]^2]))/(Cos[theta]+(Sqrt[(((nti'))^2)-Sin[theta]^2]))),1];
s[theta_]:=If[(((nti'))^2)-Sin[theta]^2>0,(((((nti'))^2) Cos[theta]-(Sqrt[(((nti'))^2)-Sin[theta]^2]))/(Cos[theta] (((nti'))^2)+(Sqrt[(((nti'))^2)-Sin[theta]^2]))),1];
plot2 = 
      Plot[{q[theta], s[theta]}, {theta, 0, (π/2)}, Frame -> True, 
        FrameTicks -> {{0, (π/4), (π/2)}, {-1, -0.5, 0, 
              0.5, 1}}, PlotRange -> {{0, (π/2)}, {-1, 1}}, 
        DisplayFunction -> Identity]; 
Show[plot2,\(Graphics[
          Text["(Subscript[r, perp])", {π\/8, 0.5}]]\),\(Graphics[
          Text["(Subscript[r, para])", {π\/8, \(-0.5\)}]]\),Graphics[Text["\!\(\* StyleBox[\(θ\_c\),\nFontSize->14]\)",{ArcSin[nti'],1.1}]],Graphics[Text["\!\(\* StyleBox[\(θ\_p\),\nFontSize->14]\)",{ArcTan[nti'],-0.2}]],DisplayFunction -> $DisplayFunction]

Again, (Subscript[r, para]) goes to zero at a certain (Subscript[θ, i]).  However, more notably, both coefficients go to a value of 1 at a certain angle.  A plot of the transmission coefficients would show, as you might expect, that both of these go to zero at the same angle.  Hence, the entirety of the light is reflected; none is transmitted.  This phenomenon is known as total internal reflection, and the angle at which it occurs is known as the critical angle.

Lets do the transmission  coefficients to complete our understanding. In the following, we plot from -.5 to 3. We also change
PlotPoints to 5000 to improve the resolution near the critical angle. For n=1.5, one of the functions approaches 2 and the other 3. Be patient -there will be a longer delay than before since Mathematica must calculate more values!

nti'=(1/n);
u[theta_]:=If[(((nti'))^2)-Sin[theta]^2>0,((2 Cos[theta])/(Cos[theta]+(Sqrt[(((nti'))^2)-Sin[theta]^2]))),0];
w[theta_]:=If[(((nti'))^2)-Sin[theta]^2>0,((2 nti' Cos[theta])/(Cos[theta] (((nti'))^2)+(Sqrt[(((nti'))^2)-Sin[theta]^2]))),0];
plot3 = 
      Plot[{u[theta], w[theta]}, {theta, 0, (π/2)}, Frame -> True, 
        FrameTicks -> {{(π/8), (π/4), ((3 π)/8)}, {-0.5`, 1, 2, 3}}, PlotPoints -> 5000, 
        PlotRange -> {{(π/8), ((3 π)/8)}, {-0.5`, 3}}, 
        DisplayFunction -> Identity]; 
Show[plot3,\(Graphics[
          Text["(Subscript[t, perp])", {\(1.5\ π\)\/8, 0.75}]]\),\(Graphics[
          Text["(Subscript[t, para])", {\(1.5\ π\)\/8, 2.0}]]\),Graphics[Text["\!\(\* StyleBox[\(θ\_c\),\nFontSize->14]\)",{ArcSin[nti'],-0.2}]],Graphics[Text["\!\(\* StyleBox[\(θ\_p\),\nFontSize->14]\)",{ArcTan[nti'],-0.2}]],DisplayFunction -> $DisplayFunction]

Even with these changes, we still do not see all of the details near  the critical angle.

Phase Shifts

Perpendicular Light

As we mentioned earlier, for a beam incident in air on a dielectric interface at nearly normal incidence the reflection coefficient of perpendicularly polarized light is equal in magnitude to that of the parallel light but opposite in sign.  This sign change corresponds with a phase shift of 180 degrees or pi radians.  In general, at any interface where the incident medium has a lower index of refraction than the transmitting medium, the perpendicular component of the reflected beam undergoes a phase change of pi radians.  When the light is incident in the dielectric, the reflected beam undergoes no phase change until the incident angle is greater than the critical angle.  At this point, the reflected beam begins to have a phase difference which grows to pi radians at an incident angle of pi/2 (glancing incidence).  Activate the following code cell to see a plot of phase shift in (Subscript[r, perp]) versus angle of incidence for internal reflection.  Note that (Subscript[θ, c]) marks the critical angle.

pShift[theta_]=If[Sin[theta]^2-(((nti'))^2)>0,-2 ArcTan[-((Sqrt[Sin[theta]^2-(((nti'))^2)])/(Cos[
                      theta]))],0];
\(plot4 = 
        Plot[pShift[theta], {theta, 0, π\/2}, 
          Frame -> 
            True, 
			FrameTicks -> {{0, π\/4, π\/2}, {0, π\/2, π}}, 
          PlotRange -> {{0, π\/2}, {0, π}}, 
			DisplayFunction -> Identity];\)
Show[plot4,Graphics[Text["(Subscript[θ, c])",{ArcSin[nti'],-0.1}]],
				DisplayFunction -> $DisplayFunction]

Parallel Light

You may also remember from the first plot that, for light incident in air on a dielectric, the parallel component of the reflected beam went to zero at the polarization angle, then took on negative values.  It stands to reason then, that we say that this component undergoes no phase shift at (Subscript[θ, i]) less than the polarization angle, and undergoes a phase shift equal to pi radians above the polarization angle.  Again, the situation is more complicated for light incident in the dielectric.  Up to the polarization angle, (Subscript[r, para]) is 180 degrees out of phase with the incident beam.  Between the polarization angle and the critical angle, there is no phase change.  Then, as with the perpendicular component, a phase change again appears after the critical angle and increases to a full 180 degrees at glancing incidence.  Activate the following code cell to see a plot of the phase change in (Subscript[r, para]) versus incident angle (for internal reflection)
.

c[theta_]=If[Sin[theta]^2-(((nti'))^2)>0,-2 ArcTan[-((Sqrt[Sin[theta]^2-(((nti'))^2)])/((((nti'))^2) Cos[theta]))],0];
d[theta_]=If[theta<ArcTan[nti'],π,c[theta]];
\(plot5 = 
        Plot[d[theta], {theta, 0, π\/2}, 
          Frame -> 
            True, 
			FrameTicks -> {{0, π\/4, π\/2}, {0, π\/2, π}}, 
			PlotRange -> {{0, π\/2}, {0, π}}, DisplayFunction -> Identity];\)
Show[plot5,Graphics[Text["(Subscript[θ, p])",{ArcTan[nti'],-0.1}]],
				Graphics[Text["(Subscript[θ, c])",{ArcSin[nti'],-0.1}]],
				DisplayFunction -> $DisplayFunction]

Note that, for this plot, (Subscript[θ, p]) marks the polarization angle, while (Subscript[θ, c]) again marks the location of the critical angle.  In case you were wondering, the transmitted beams undergo no phase shift in either component, in either case of incidence.

Relative Phase Shift

One final plot that is of interest is that of the relative phase shift between the perpendicular and parallel components of the internally reflected beam.  This behaviour is useful for producing certain types of polarizing optics. Activate the following code cell to see this plot.

\(plot6 = 
        Plot[d[theta] - pShift[theta], {theta, 0, π\/2}, 
          Frame -> 
            True, 
				FrameTicks -> {{0, π\/4, π\/2}, {0, π\/2, π}}, 
				PlotRange -> {{0, π\/2}, {0, π}}, 
          DisplayFunction -> Identity];\)
Show[plot6,Graphics[Text["(Subscript[θ, p])",{ArcTan[nti'],-0.1}]],
					Graphics[Text["(Subscript[θ, c])",{ArcSin[nti'],-0.1}]],
					DisplayFunction -> $DisplayFunction]

Reflectance and Transmittance [External]

We have defined our four coefficients in terms of the relative amplitudes of the incident, transmitted, and reflected E-fields.  But what if we want to know how much of the incident energy is reflected or transmitted?  The energy carried by a light beam is proportional to the square of the amplitude of its E-field, but also depends on the cross-sectional area of the beam.  The cross-sectional area of each beam is a function of the angle that beam makes with the surface of the interface, or the normal to it.  So, since (Subscript[θ, r]) = (Subscript[θ, i]), the cross-sectional areas of the incident and reflected beams are equal and we can define the relative flux density of these two beams simply in terms of their field amplitudes squared.  We call this dimensionless quantity "reflectance".

R = (r^2)

The angle of the transmitted beam, (Subscript[θ, t]) does not equal (Subscript[θ, i]), so an additional factor must be added to the expression for the complementary quantity known as "transmittance".

T = (((Subscript[n, t]) cos(Subscript[θ, t]))/((Subscript[n, i]) cos(Subscript[θ, i]))) 
          (t^2) = (((Subscript[n, ti]) cos(Subscript[θ, t]))/cos)(Subscript[θ, i]) 
          (t^2)

It can be shown that R+T=1 under any circumstances, conservation of energy  dictate this.

R + T = 1

These formulations of R and T indicate that we can also resolve these two quantities into orthogonal polarization components simply by substituting that component of r or t into the expression for R or T.  Activating the following two code cells will bring up plots of the perpendicular and parallel components of R and T, respectively, plotted against (Subscript[θ, i]).

\(plot7 = 
      Plot[{rPerp[theta]\^2, \(\@\(nti\^2 - Sin[theta]\^2\)\ tPerp[theta]\^2\)\/Cos[theta]}, {theta, 0, π\/2}, Frame -> True, 
        FrameTicks -> {{0, π\/4, π\/2}, {\(-1\), \(-0.5\), 0, 
              0.5, 1}}, PlotRange -> {{0, π\/2}, {0, 1}}, 
        PlotLabel -> "R and T (perpendicular) vs. (Subscript[θ, i])", 
        DisplayFunction -> Identity]; \)
\(Show[plot7, 
      Graphics[Text["(Subscript[T, perp])", {π\/5, 0.75}]], 
      Graphics[Text["(Subscript[R, perp])", {π\/5, 0.25}]], DisplayFunction -> $DisplayFunction]\)

And now the Parallel case!

\(plot8 = 
      Plot[{rPara[theta]\^2, \(\@\(nti\^2 - Sin[theta]\^2\)\ tPara[theta]\^2\)\/Cos[theta]}, {theta, 0, π\/2}, Frame -> True, 
        FrameTicks -> {{0, π\/4, π\/2}, {\(-1\), \(-0.5\), 0, 
              0.5, 1}}, PlotRange -> {{0, π\/2}, {0, 1}}, 
        PlotLabel -> "R and T (parallel) vs. (Subscript[θ, i])", 
        DisplayFunction -> Identity]; \)
\(Show[plot8, 
      Graphics[Text["(Subscript[T, para])", {π\/5, 0.9}]], 
      Graphics[Text["(Subscript[R, para])", {π\/5, 0.1}]], DisplayFunction -> $DisplayFunction]\)

Reflectance and Transmittance At Normal Incidence [External]

It is interesting to note that at (Subscript[θ, i]) = 0, the value of R does not go to zero.  In other words, there is some finite portion of the incident beam refracted even at normal incidence.  If we substitute (Subscript[θ, i]) = 0 into the expression for either component of R and T, we get.

\(\(\(R = \(R\_para = \(R\_perp = \(\((\(n\_t - n\_i\)\/\(n\_t + n\_i\))\)\^2 = \((\(n\_ti - 1\)\/\(n\_ti + 1\))\)\^2\)\)\)\)\(
\)
    \)\)
\(T = \(T\_para = \(\(T\_perp = \(4 \( n\_t\) n\_i\)\/\((n\_t + n\_i)\)\^2\)\(
\)
        \)\)\)

So at higher indices of refraction, there will be more reflection at normal incidence.  To see a plot of this dependence, activate the following code cell:

Plot[(((((n - 1)/(n + 1))))^2), {n, 1, 4}, Frame -> True, 
      FrameTicks -> {{1, 2, 3, 4}, {0.2, 0.4}}, 
      PlotRange -> {{1, 4}, {0, 0.5`}}, 
      FrameLabel -> {"\!\(\* StyleBox[\"refractive\",\nFontSize->14]\)\
\!\(\* StyleBox[\" \",\nFontSize->14]\)\!\(\* StyleBox[\"index\",\n\
FontSize->14]\)", R}]

Reflectances For Internal Reflection

nti'=(1/n);
x[theta_]:=If[(((nti'))^2)-Sin[theta]^2>0,(((((Cos[theta]-(Sqrt[(((nti'))^2)-Sin[theta]^2]))/(Cos[theta]+(Sqrt[(((nti'))^2)-Sin[theta]^2])))))^2),1];
y[theta_]:=If[(((nti'))^2)-Sin[theta]^2>0,((((((((nti'))^2) Cos[theta]-(Sqrt[(((nti'))^2)-Sin[theta]^2]))/(Cos[theta] (((nti'))^2)+(Sqrt[(((nti'))^2)-Sin[theta]^2])))))^2),1];
plot9 = 
      Plot[{x[theta], y[theta]}, {theta, 0, (π/2)}, Frame -> True, 
        FrameTicks -> {{0, (π/4), (π/2)}, {0, 
              0.1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}}, 
        PlotPoints -> 1000, 
        PlotLabel -> "\!\(\* StyleBox[\"Reflectances\",\nFontSize->14]\
\)\!\(\* StyleBox[\"-\",\nFontSize->14]\)\!\(\* StyleBox[\"Internal\",\n\
FontSize->14]\)", PlotRange -> {{0, (π/2)}, {0, 1}}, 
        DisplayFunction -> Identity]; 
Show[plot9,\(Graphics[
          Text["(Subscript[R, perp])", {π\/8, 0.25}]]\),\(Graphics[
          Text["(Subscript[R, para])", {π\/3.5, 0.25}]]\),Graphics[Text["\!\(\* StyleBox[\(θ\_c\),\nFontSize->14]\)",{ArcSin[nti'],-0.1}]],Graphics[Text["\!\(\* StyleBox[\(θ\_p\),\nFontSize->14]\)",{ArcTan[nti'],-0.1}]],DisplayFunction -> $DisplayFunction]

More on The Critical Angle

Let's look at the critical angle as a function of index for substances from water to diamond (1.33-2.5) with air on one side.

(Subscript[θ, c]) = ArcSin (1/n)

Take a few minutes to see if you can sketch this function by hand. Then let Mathematica do it.

θc[n_] := ArcSin[1/n];

Plot[θc[n], {n, 1.3`, 2.5`}, Frame -> True, 
      FrameTicks -> {{1.3`, 1.5`, 1.7`, 1.9`, 2.1`, 2.3`, 2.5`}, {0, 
            10 °, 20 °, 30 °, 40 °, 
            50 °, 60 °}}, 
      PlotLabel -> "\!\(\* StyleBox[\"Critical\",\nFontSize->14]\)\!\(\
\* StyleBox[\" \",\nFontSize->14]\)\!\(\* StyleBox[\"Angle\",\n\
FontSize->14]\)\!\(\* StyleBox[\" \",\nFontSize->14]\)\!\(\* StyleBox[\"vs\",\
\nFontSize->14]\)\!\(\* StyleBox[\" \",\nFontSize->14]\)\!\(\* \
StyleBox[\"Index\",\nFontSize->14]\)", 
      PlotRange -> {{1.3`, 2.5`}, {0, 60 °}}]

More on The Polarizing Angle (Brewster's Angle)

Let's look at the polarizing angle as a function of index for substances from water to diamond (1.33-2.5) with air on one side. We can do this for external and internal reflection.

External Reflection

(Subscript[θ, p]) = ArcTan (n)

Take a few minutes to see if you can sketch this function by hand. Then let Mathematica do it.

θpext[n_] := ArcTan[n];

Plot[θpext[n], {n, 1.3`, 2.5`}, Frame -> True, 
      FrameTicks -> {{1.3`, 1.5, 1.7, 1.9, 2.1, 2.3, 
            2.5`}, {50 °, 55 °, 60 °, 
            65 °, 70 °}}, 
      PlotLabel -> "\!\(\* StyleBox[\"Polarizing\",\nFontSize->14]\)\!\
\(\* StyleBox[\" \",\nFontSize->14]\)\!\(\* StyleBox[\"Angle\",\n\
FontSize->14]\)\!\(\* StyleBox[\" \",\nFontSize->14]\)\!\(\* StyleBox[\"vs\",\
\nFontSize->14]\)\!\(\* StyleBox[\" \",\nFontSize->14]\)\!\(\* \
StyleBox[\"Index\",\nFontSize->14]\)", 
      PlotRange -> {{1.3`, 2.5`}, {50 °, 
            70 °}}]

Internal Reflection

(Subscript[θ, p]) = ArcTan (1/n)

Take a few minutes to see if you can sketch this function by hand. Then let Mathematica do it.

θpint[n_] := ArcTan[1/n];

Plot[θpint[n], {n, 1.3`, 2.5`}, Frame -> True, 
      FrameTicks -> {{1.3`, 1.5, 1.7, 1.9, 2.1, 2.3, 
            2.5`}, {20 °, 25 °, 30 °, 
            35 °, 40 °}}, 
      PlotLabel -> "\!\(\* StyleBox[\"Polarizing\",\nFontSize->14]\)\!\
\(\* StyleBox[\" \",\nFontSize->14]\)\!\(\* StyleBox[\"Angle\",\n\
FontSize->14]\)\!\(\* StyleBox[\" \",\nFontSize->14]\)\!\(\* StyleBox[\"vs\",\
\nFontSize->14]\)\!\(\* StyleBox[\" \",\nFontSize->14]\)\!\(\* \
StyleBox[\"Index\",\nFontSize->14]\)", 
      PlotRange -> {{1.3`, 2.5`}, {20 °, 
            40 °}}]

References

Hecht, E., OPTICS, Addison-Wesley, Reading, Mass., 1998.