Polarization
Polarization describes how the electric-field vector of a transverse wave changes in the plane perpendicular to propagation. It is not the path of a photon, the direction the whole wave travels, or merely a color used to draw the field. It is a measurable property that affects transmission through polarizers, reflection, scattering, and birefringent materials.
At one fixed position in a monochromatic wave, two perpendicular electric-field components can trace a line, circle, or ellipse as time passes. Linear and circular polarization are special cases of the general polarization ellipse.
Two transverse components
For a wave traveling in positive z, choose transverse x and y basis vectors. At a fixed position, write
\[E_x(t)=A_x\cos(\omega t),\qquad E_y(t)=A_y\cos(\omega t+\delta)\]The amplitudes \(A_x\) and \(A_y\) are nonnegative choices of magnitude. The relative phase \(\delta\) contains information that intensity in each component alone cannot recover. Adding the same phase to both components changes the time origin but not the polarization shape.
import numpy as np
import matplotlib.pyplot as plt
phase = np.linspace(0, 2 * np.pi, 361)
x_amplitude, y_amplitude, relative_phase = 1.0, 0.5, np.pi / 2
ex = x_amplitude * np.cos(phase)
ey = y_amplitude * np.cos(phase + relative_phase)
fig, ax = plt.subplots()
ax.plot(ex, ey)
ax.set(xlabel="Ex (V/m)", ylabel="Ey (V/m)")
ax.set_aspect("equal")
plt.show()
The two components must be sampled at the same position and time. Combining values from different timestamps can fabricate a false ellipse.
Linear polarization
If the components are in phase or opposite in phase,
\[\delta=0\quad\text{or}\quad\delta=\pi\]the electric-field tip moves back and forth along a fixed line. A zero amplitude in one component is also linear polarization along the remaining basis axis.
For in-phase components, the line angle \(\psi\) relative to positive x satisfies
\[\tan\psi=\frac{A_y}{A_x}\]For a phase difference of pi, one component reverses and the line has the corresponding opposite slope. The electric field changes sign every half-cycle, but the axis of linear polarization is unchanged.
Circular polarization
Circular polarization requires equal nonzero amplitudes and a quarter-cycle relative phase:
\[A_x=A_y=A,qquad \delta=\pm\frac{\pi}{2}\]Then
\[E_x^2+E_y^2=A^2\]The electric-field tip rotates at constant radius. The sign of the relative phase sets the direction of rotation, but the words right-handed and left-handed depend on whether the observer looks along or against the propagation direction and on the convention used by the reference. Always state the viewing direction instead of relying on a label alone.
Elliptical polarization
Most other combinations of nonzero amplitudes and relative phase produce an ellipse. Eliminating time from the component equations gives
\[\left(\frac{E_x}{A_x}\right)^2 +\left(\frac{E_y}{A_y}\right)^2 -2\frac{E_xE_y}{A_xA_y}\cos\delta =\sin^2\delta\]This equation is difficult to read as a classification rule, so use the limiting cases:
if x_amplitude == 0 or y_amplitude == 0 or np.isclose(np.sin(relative_phase), 0):
kind = "linear"
elif np.isclose(x_amplitude, y_amplitude) and np.isclose(np.cos(relative_phase), 0):
kind = "circular"
else:
kind = "elliptical"
print("Polarization:", kind)
Numerical comparisons need tolerances because measured and floating-point values rarely equal the ideal conditions exactly. The tolerance should be reported rather than hidden.
Ideal linear polarizer
An ideal linear polarizer transmits the electric-field component along its transmission axis. If \(\hat{\mathbf a}\) is a unit vector, the transmitted field is the projection
\[\mathbf E_{\mathrm{out}} =\left(\mathbf E_{\mathrm{in}}\cdot\hat{\mathbf a}\right) \hat{\mathbf a}\]For incident linear polarization at angle \(\theta\), the field amplitude is multiplied by \(\cos\theta\). Because intensity is proportional to squared field amplitude, Malus's law is
\[I=I_0\cos^2\theta\]Parallel ideal polarizers transmit the same selected linear component. Crossed ideal polarizers have \(\theta=90^\circ\) and transmit zero intensity in the ideal model. Real polarizers have finite extinction, absorption, wavelength dependence, and angular imperfections.
Unpolarized incident light
Unpolarized light has no fixed long-term transverse direction or relative phase. An ideal linear polarizer transmits half its average intensity:
\[I_{\mathrm{after\ first}}=\frac{1}{2}I_{\mathrm{unpolarized}}\]A second ideal polarizer at angle \(\theta\) to the first then gives
\[I=\frac{1}{2}I_{\mathrm{unpolarized}}\cos^2\theta\]Do not apply the one-half factor repeatedly as if light becomes unpolarized after each polarizer. After the first ideal polarizer, the transmitted light is linearly polarized.
Three-polarizer result
Putting a third polarizer between crossed polarizers can increase transmission. If the first and last axes differ by ninety degrees and the middle axis is at forty-five degrees, then for initially unpolarized light
\[I_{\mathrm{out}} =\frac{I_0}{2} \cos^2(45^\circ) \cos^2(45^\circ) =\frac{I_0}{8}\]The middle element changes the polarization state before the last projection. It does not add energy; each ideal projection can only preserve or reduce intensity.
Wave plates
A birefringent wave plate gives two orthogonal field components different phase delays. A quarter-wave plate introduces a relative phase shift of \(\pi/2\). Linear polarization with equal components along its fast and slow axes can therefore become circular. Other input angles produce elliptical polarization.
A half-wave plate introduces a relative phase shift of \(\pi\). It can rotate the axis of linear polarization. Wave plates ideally change relative phase rather than simply discarding the perpendicular component as a polarizer does.
Jones-vector representation
For fully polarized monochromatic light, a complex two-component vector can store amplitude and phase compactly:
\[\mathbf J= \begin{pmatrix} A_x\\ A_ye^{i\delta} \end{pmatrix}\]An overall complex phase does not change the polarization ellipse. Matrices can represent ideal polarizers, wave plates, and rotations. The complex notation is bookkeeping for relative phase; the physical electric field is real.
Jones vectors do not describe partially polarized or unpolarized light by themselves. Stokes parameters or coherence matrices are used when statistical polarization state is important.
Polarization by reflection and scattering
Reflection coefficients generally differ for electric fields parallel and perpendicular to the plane of incidence. At Brewster's angle for ideal nonmagnetic transparent media, the reflected parallel component vanishes:
\[\tan\theta_B=\frac{n_2}{n_1}\]The reflected light is then linearly polarized perpendicular to the plane of incidence when the incident light contains both components. Scattering can also preferentially select polarization directions, which helps explain polarization patterns in skylight.
Verification and limits
Set one component to zero and verify a line. Use equal amplitudes and a quarter-cycle phase difference and verify constant squared magnitude. Add a common phase to both components and verify the same ellipse. For a polarizer, verify parallel transmission, zero ideal crossed transmission, and Malus's cosine-squared dependence.
This treatment assumes a monochromatic transverse wave and ideal optical elements. Real light can be broadband, partially polarized, spatially varying, incoherent, scattered, or depolarized. Materials can absorb, disperse, and mix polarization across a beam. Polarization is a field state and measurement probability structure; the ellipse is not a literal trajectory followed by an individual photon.
A worked Python notebook
Pair this component with a first component equal to cosine of phase to form an ellipse. The question for this calculation is precise: how does second transverse field component change when we change phase, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.
Polarization requires both components and their relative phase, not one component alone. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 6.28319 rad. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.
Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.
State the relationship before choosing a graph
In this small example, q stands for phase, expressed in rad. The output is
second transverse field component, expressed in V/m. The equation below uses the numerical values in those
units. Any fixed coefficients therefore carry the units needed to make the relation
consistent. A value supplied in another unit must be converted before it enters this
expression.
This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.
At 0.785398 rad, the expression gives -0.353553 V/m. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.
import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
q = sp.Symbol("q", real=True)
expression = sp.sympify("0.5*cos(q+pi/2)", locals={"q": q})
reference_input = 0.7853981633974483
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "rad")
print("Predicted second transverse field component:", reference_output, "V/m")
sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement
to that symbol. subs then replaces the symbol with our chosen input, and float
converts the resulting symbolic number into a standard Python numerical value.
Keeping these steps separate lets us inspect the equation before numerical substitution.
It also prevents a later plot from becoming the only surviving record of the model.
Compare a few cases you can explain
Before drawing hundreds of points, calculate three. The lower endpoint gives 0 V/m. The midpoint, at 3.14159 rad, gives -6.12323e-17 V/m. The upper endpoint gives 1.22465e-16 V/m. Read these together with the assumptions: they describe different choices of phase within the same example, with the other inputs held fixed.
These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.
evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 3.141592653589793, 6.283185307179586])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
print(f"{input_value:.6g} rad -> {output_value:.6g} V/m")
assert np.all(np.isfinite(outputs))
lambdify turns the symbolic expression into a numerical function. We explicitly select
NumPy as its numerical backend so it can accept an array. The same expression now has
two useful forms: one for algebra and one for calculating many cases. The assertion
checks that the chosen inputs produce finite outputs. It would catch an infinity or an
undefined numerical result, but a finite value can still be physically inappropriate.
Make a graph that answers the question
The horizontal axis shows phase; the vertical axis shows second transverse field component. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.
input_grid = np.linspace(0.0, 6.283185307179586, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Phase (rad)")
ax.set_ylabel("Second transverse field component (V/m)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()
Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.
Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.
Measure the local sensitivity
A derivative asks a more specific question than the whole graph: how quickly does second transverse field component change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.785398 rad is -0.353553 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.
\[f'(q) = - 0.5 \cos{\left(q \right)}\]For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.
slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.8168140899333463
linear_prediction = reference_output + slope_at_reference * (
changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "V/m")
print("Full expression:", full_prediction, "V/m")
print("Approximation difference:", full_prediction - linear_prediction, "V/m")
Here the changed input is 0.816814 rad. The tangent estimate is -0.364661 V/m, while the full expression gives -0.364484 V/m. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.
This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.
Check the derivative with numerical differences
We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.
The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.
input_span = 6.283185307179586
for fraction in [1e-2, 1e-3, 1e-4]:
step = input_span * fraction
upper = float(evaluate(reference_input + step))
lower = float(evaluate(reference_input - step))
numerical_slope = (upper - lower) / (2 * step)
difference = numerical_slope - slope_at_reference
print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")
For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.
An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.
Change the example and preserve the reasoning
Make one copy of the reference calculation before editing it. Choose a second value of phase inside the stated interval, predict whether second transverse field component will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.
Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.
Finally, return to the physical limit stated at the start: Polarization requires both components and their relative phase, not one component alone. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.
Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.
Recover the output change by integrating the slope
There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of V/m: the input unit in the integration cancels the input unit in the denominator of the slope.
This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.
slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(6.283185307179586) - evaluate(0.0))
for points in [101, 201, 401]:
grid = np.linspace(0.0, 6.283185307179586, points)
slopes = np.asarray(slope_function(grid), dtype=float)
slopes = np.broadcast_to(slopes, grid.shape)
integrated_change = np.trapezoid(slopes, grid)
print(
points,
"integrated change:",
integrated_change,
"endpoint change:",
endpoint_change,
)
The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.
Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.
Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.