Wave properties of light
Light is an electromagnetic wave with frequency, wavelength, phase, amplitude, direction, and polarization. These properties are related but not interchangeable. Frequency sets the rate of field oscillation. Wavelength sets spatial repetition. Amplitude controls field strength and, through its square, intensity. Phase locates a wave within its cycle.
The values needed in a model depend on the question. Refraction needs frequency and a material index. Interference needs relative phase and coherence. Diffraction needs wavelength and aperture geometry. Polarization needs two transverse components.
Frequency, period, wavelength, and speed
For a periodic wave,
\[f=\frac{1}{T}\]where frequency \(f\) is measured in hertz and period \(T\) in seconds. Wave speed, wavelength, and frequency obey
\[v=\lambda f\]In vacuum,
\[c=299\,792\,458\ \mathrm{m/s}\]and
\[c=\lambda_0f\]The vacuum light speed is exact in SI. The vacuum wavelength \(\lambda_0\) follows from the frequency.
A sinusoidal field and phase
A linearly polarized plane-wave component can be written
\[E(x,t)=E_0\cos(kx-\omega t+\phi_0)\]The wave number and angular frequency are
\[k=\frac{2\pi}{\lambda},\qquad \omega=2\pi f\]The phase is
\[\phi(x,t)=kx-\omega t+\phi_0\]Increasing position by one wavelength or time by one period changes phase by a whole cycle. A common phase shift changes where the cycle starts but not frequency, wavelength, amplitude, or polarization shape.
import numpy as np
c, frequency = 299792458.0, 5e14
wavelength = c / frequency
k, omega = 2 * np.pi / wavelength, 2 * np.pi * frequency
position, time = 1e-6, 1e-15
phase = k * position - omega * time
print(f"Vacuum wavelength: {wavelength*1e9:.2f} nm")
print(f"Phase: {phase:.3f} rad")
Propagation through a medium
For an ideal medium with refractive index \(n\),
\[n=\frac{c}{v_p}\]so the phase velocity and medium wavelength are
\[v_p=\frac{c}{n},\qquad \lambda=\frac{\lambda_0}{n}\]At a stationary interface, frequency stays the same because the fields on both sides must share the same temporal oscillation. Speed and wavelength change together. A photon's energy also depends on frequency, so it does not drop merely because wavelength is shorter in the medium.
Worked wavelength conversion
Take frequency \(f=6.00\times10^{14}\ \mathrm{Hz}\). Its vacuum wavelength is
\[\lambda_0=\frac{c}{f} =\frac{2.998\times10^8\ \mathrm{m/s}} {6.00\times10^{14}\ \mathrm{Hz}} \approx5.00\times10^{-7}\ \mathrm m\]This is five hundred nanometres. In an ideal medium with \(n=1.50\), the phase speed is about \(2.00\times10^8\ \mathrm{m/s}\) and the wavelength is about \(333\ \mathrm{nm}\). The frequency remains the original value.
Transverse fields
In a source-free vacuum plane wave, the electric field, magnetic field, and propagation direction are mutually perpendicular:
\[\mathbf E\perp\mathbf B, \qquad \mathbf E\perp\mathbf k, \qquad \mathbf B\perp\mathbf k\]Their amplitudes satisfy
\[E_0=cB_0\]and \(\mathbf E\times\mathbf B\) points with energy propagation. Light has two independent transverse polarization directions.
Amplitude and intensity
For a sinusoidal vacuum plane wave, average intensity is
\[I=\frac{1}{2}\varepsilon_0cE_0^2 =\varepsilon_0cE_{\mathrm{rms}}^2\]Doubling electric-field amplitude quadruples intensity. Intensity is an energy-flow rate per area, not field amplitude. The peak amplitude and root-mean-square amplitude differ by a factor of the square root of two for a sinusoid, so formulas must identify which one they use.
Electromagnetic spectrum
Radio, microwave, infrared, visible, ultraviolet, X-ray, and gamma radiation are ranges of the same electromagnetic phenomenon. Boundaries between names are conventional and overlap across applications. Frequency or vacuum wavelength is a clearer physical label than a band name.
Visible color perception depends on the spectrum reaching the eye and on biological response. Monochromatic wavelength is not the same thing as a screen RGB value. Many different spectra can produce similar perceived color, and wavelengths outside the visible range have no visible color.
The quantum energy of one photon is
\[E_{\mathrm{photon}}=hf=\frac{hc}{\lambda_0}\]Greater frequency means greater energy per photon. Classical wave intensity instead describes energy per area per time and can vary independently through photon arrival rate.
Dispersion and group velocity
Real refractive index often depends on angular frequency:
\[n=n(\omega)\]The phase velocity is
\[v_p=\frac{\omega}{k}\]while the group velocity of a narrowband wave packet is
\[v_g=\frac{d\omega}{dk}\]In a nondispersive vacuum they are equal. In a dispersive medium they can differ, causing frequency components to accumulate different phases and pulses to broaden or reshape. Neither quantity should be described casually as information speed in every strongly absorbing or anomalously dispersive situation.
Coherence
Interference requires stable relative phase. Temporal coherence describes how long a wave remains phase-correlated with a delayed copy of itself. Spatial coherence describes phase correlation across different positions on a wavefront. A perfectly monochromatic infinite plane wave is completely coherent but unphysical; real sources have finite bandwidth, duration, and extent.
A rough narrowband coherence length scale is
\[L_c\sim\frac{c}{\Delta f}\]The numerical factor and detailed shape depend on how bandwidth and coherence are defined.
Polarization
Polarization records the relative amplitudes and phase of two transverse electric-field components. Linear polarization traces a line, circular polarization traces a circle, and the general state traces an ellipse. Unpolarized or partially polarized light requires statistical information rather than one deterministic field pair.
Polarization is independent of frequency in the sense that waves of many frequencies can share a polarization state, but real optical elements transform polarization differently across wavelength.
Wave packets and finite signals
A sinusoidal wave extends forever and carries no localized message. A finite pulse is a superposition of frequencies. Narrowing a pulse in time generally requires broader frequency bandwidth. The phase relationships among those frequencies determine the pulse shape.
A pulse can also have a carrier frequency and a slowly varying envelope. Sampling must resolve the carrier if the instantaneous field matters, or use a justified envelope model when only slower behavior matters.
Verification and limits
Check units in \(v=\lambda f\). At fixed frequency, lowering phase speed must shorten wavelength. Advancing one wavelength or one period must change phase by a full cycle. For a vacuum plane wave, verify transverse fields, the amplitude ratio, and energy-flow direction.
The simple relations assume linear propagation and clearly defined frequency. Real media can absorb, scatter, disperse, be anisotropic, move, or respond nonlinearly. Very short pulses need broadband models. Classical waves predict field propagation and interference; quantum models are needed for discrete photon emission, absorption, and detection. The two descriptions agree on many measurable averages but answer different questions.
A worked Python notebook
Convert wavelengths across a visible-light example range into frequencies. The question for this calculation is precise: how does frequency change when we change vacuum wavelength, 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.
Wavelength changes on entering a medium, while frequency is set by the source and interface conditions. 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 4e-07 to 7e-07 m. 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 vacuum wavelength, expressed in m. The output is
frequency, expressed in Hz. 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 5e-07 m, the expression gives 5.99585e+14 Hz. 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("299792458/q", locals={"q": q})
reference_input = 5e-07
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted frequency:", reference_output, "Hz")
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 7.49481e+14 Hz. The midpoint, at 5.5e-07 m, gives 5.45077e+14 Hz. The upper endpoint gives 4.28275e+14 Hz. Read these together with the assumptions: they describe different choices of vacuum wavelength 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([4e-07, 5.5e-07, 7e-07])
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} m -> {output_value:.6g} Hz")
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 vacuum wavelength; the vertical axis shows frequency. 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(4e-07, 7e-07, 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("Vacuum wavelength (m)")
ax.set_ylabel("Frequency (Hz)")
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 frequency change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 5e-07 m is -1.19917e+21 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.
\[f'(q) = - \frac{299792458}{q^{2}}\]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 = 5.015e-07
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, "Hz")
print("Full expression:", full_prediction, "Hz")
print("Approximation difference:", full_prediction - linear_prediction, "Hz")
Here the changed input is 5.015e-07 m. The tangent estimate is 5.97786e+14 Hz, while the full expression gives 5.97792e+14 Hz. 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 = 3e-07
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 vacuum wavelength inside the stated interval, predict whether frequency 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: Wavelength changes on entering a medium, while frequency is set by the source and interface conditions. 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 Hz: 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(7e-07) - evaluate(4e-07))
for points in [101, 201, 401]:
grid = np.linspace(4e-07, 7e-07, 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.