Reflection and refraction
When light reaches a boundary between materials, some may reflect and some may transmit. The transmitted wave usually changes speed and direction; this change of direction is refraction. Ray optics represents the wave by narrow paths and works when the relevant surfaces and apertures are large compared with wavelength.
Every angle in the standard laws is measured from the normal, an imaginary line perpendicular to the interface. Measuring from the surface produces complementary angles and is one of the most common sources of incorrect ray diagrams.
Refractive index
The refractive index of a transparent medium is
\[n=\frac{c}{v}\]Here \(c\) is the speed of light in vacuum and \(v\) is the phase speed at the specified frequency. Ordinary transparent materials usually have \(n>1\) in visible light.
Frequency remains continuous across a stationary boundary because the fields on both sides must oscillate together. Since \(v=f\lambda\), the wavelength changes:
\[\lambda_{\mathrm{medium}}=\frac{\lambda_0}{n}\]The vacuum wavelength is \(\lambda_0\). In a dispersive medium, refractive index depends on wavelength or frequency, so the relation must use the appropriate value of \(n\).
Law of reflection
The incident ray, reflected ray, and surface normal lie in one plane. Their angles obey from the normal satisfy
\[\theta_r=\theta_i\]Reflection does not require the second material to be a mirror. Any optical interface can reflect some of the wave. The amount depends on refractive indices, incidence angle, polarization, absorption, and surface coatings; the equality above determines direction, not reflected intensity.
For unit incident direction \(\hat{\mathbf d}\) pointing toward an interface and a consistently oriented unit normal \(\hat{\mathbf n}\), the reflected direction is
\[\hat{\mathbf d}_r =\hat{\mathbf d} -2(\hat{\mathbf d}\cdot\hat{\mathbf n}) \hat{\mathbf n}\]Reversing the normal leaves this reflected direction unchanged because the normal appears twice. The refraction calculation must additionally know which medium is incident.
Snell's law
For refraction from medium 1 into medium 2,
\[n_1\sin\theta_1=n_2\sin\theta_2\]If \(n_2>n_1\), then the transmitted angle is smaller: the ray bends toward the normal. If \(n_2<n_1\), the transmitted angle is larger: the ray bends away from the normal.
At normal incidence, \(\theta_1=0\) and the transmitted ray continues without a directional bend. Its speed and wavelength can still change, and partial reflection can still occur.
Worked air-to-glass example
Let light pass from air with \(n_1=1.00\) into glass with \(n_2=1.50\) at incident angle \(\theta_1=30^\circ\). Then
\[\sin\theta_2 =\frac{n_1}{n_2}\sin\theta_1 =\frac{1.00}{1.50}\sin30^\circ =\frac{1}{3}\]Therefore
\[\theta_2=\sin^{-1}\left(\frac{1}{3}\right) \approx19.5^\circ\]The smaller angle confirms the predicted bend toward the normal.
Total internal reflection
When light travels from higher index to lower index, Snell's law may request a sine larger than one. Then no propagating transmitted ray exists and the ideal lossless interface produces total internal reflection.
The critical angle occurs when the transmitted ray would make ninety degrees with the normal:
\[\sin\theta_c=\frac{n_2}{n_1}, \qquad n_1>n_2\]For incidence angles greater than \(\theta_c\), total internal reflection occurs. At the boundary, an evanescent field still extends into the lower-index medium and decays rapidly; “no transmitted ray” does not mean the electromagnetic field is exactly zero immediately beyond the surface.
import numpy as np
n_incident, n_transmitted = 1.5, 1.0
for angle_degrees in [20, 40, 50]:
sine_transmitted = n_incident / n_transmitted * np.sin(np.deg2rad(angle_degrees))
if abs(sine_transmitted) > 1:
print(angle_degrees, "degrees: total internal reflection")
else:
angle = np.rad2deg(np.arcsin(sine_transmitted))
print(angle_degrees, f"degrees: transmitted angle {angle:.3f} degrees")
Floating-point roundoff near the critical angle needs a documented tolerance. A value barely above one because of rounding may be clamped only when it is within that tolerance; a physically larger value must return the reflection branch.
Reversibility
Snell's law is reversible for ordinary stationary reciprocal media. If a ray travels along a refracted path and its direction is reversed, it retraces the original incident path. This gives a strong check: swap the medium indices and use the former transmitted angle as the new incidence angle. The recovered angle should equal the original one.
Reversibility does not say transmitted intensity is always the same in an arbitrary measurement. Beam area, impedance, polarization, absorption, and how power is normalized must also be considered.
Why refraction occurs
The tangential phase of a wave must match along the interface. That requirement preserves the component of wave vector parallel to the boundary:
\[k_1\sin\theta_1=k_2\sin\theta_2\]Using \(k=n\omega/c\) recovers Snell's law. This wave explanation shows that refraction is not caused by one side of a photon mechanically slowing first. The ray is a geometric description of a wave whose phase obeys boundary conditions.
Fermat's principle gives another equivalent viewpoint: the physical ray makes travel time stationary with respect to nearby paths. Applying that principle to a planar boundary also produces Snell's law.
Dispersion
Refractive index often varies with wavelength:
\[n=n(\lambda)\]Different colors therefore refract through different angles. This produces prism dispersion and chromatic aberration in simple lenses. In normal dispersion across much of the visible range, shorter wavelengths commonly have larger refractive index, but the actual material data and wavelength range determine the trend.
Phase velocity \(v_p=c/n\) and the speed of a pulse envelope need not be identical in a dispersive medium. A single refractive index is insufficient for broadband pulses.
Reflection and transmitted power
The direction laws do not determine how much power reflects or transmits. Fresnel coefficients depend on polarization and angle. At normal incidence between nonmagnetic, lossless media, the reflected intensity fraction is
\[R=\left(\frac{n_1-n_2}{n_1+n_2}\right)^2\]The corresponding transmitted energy fraction is \(T=1-R\) when defined as a power ratio with the proper impedance factor already included. Squaring a field-amplitude coefficient without that factor can give an incorrect transmitted power.
Multiple parallel interfaces
For a flat slab with parallel faces, the ray bends at the first surface and bends back at the second. When the exterior medium is the same on both sides, the emerging ray is parallel to the incident ray but laterally displaced. If the faces are not parallel, as in a prism, the net direction changes.
Ray tracing through several surfaces must update the current medium after every successful intersection and refraction. Using the original incident index at later surfaces produces plausible but wrong paths.
Verification and limits
Check equal incidence and reflection angles. Equal refractive indices must preserve the ray direction. Air-to-glass refraction must bend toward the normal, while glass-to-air refraction bends away until the critical angle. Reverse a ray and confirm it retraces the path. Reject invalid inverse-sine arguments as total internal reflection.
Ray optics omits diffraction, interference, finite beam width, coherence, near fields, surface roughness, and wavelength-scale structures. Ideal formulas also omit absorption, anisotropy, nonlinear response, moving interfaces, and polarization-dependent power unless those are added explicitly. Near an aperture or feature comparable with wavelength, use wave optics rather than forcing a ray model to answer the wrong question.
A worked Python notebook
Refract a ray from air into glass with refractive index 1.5. The question for this calculation is precise: how does transmitted angle change when we change incident angle, 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.
Angles are measured from the normal and the indices are fixed at the chosen wavelength. 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 1.5708 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 incident angle, expressed in rad. The output is
transmitted angle, expressed in rad. 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 0.490883 rad. 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("asin(sin(q)/1.5)", locals={"q": q})
reference_input = 0.7853981633974483
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "rad")
print("Predicted transmitted angle:", reference_output, "rad")
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 rad. The midpoint, at 0.785398 rad, gives 0.490883 rad. The upper endpoint gives 0.729728 rad. Read these together with the assumptions: they describe different choices of incident angle 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, 0.7853981633974483, 1.5707963267948966])
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} rad")
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 incident angle; the vertical axis shows transmitted angle. 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, 1.5707963267948966, 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("Incident angle (rad)")
ax.set_ylabel("Transmitted angle (rad)")
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 transmitted angle 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 0.534522 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.
\[f'(q) = \frac{0.666666666666667 \cos{\left(q \right)}}{\sqrt{1 - 0.444444444444444 \sin^{2}{\left(q \right)}}}\]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 = 0.7932521450314227
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, "rad")
print("Full expression:", full_prediction, "rad")
print("Approximation difference:", full_prediction - linear_prediction, "rad")
Here the changed input is 0.793252 rad. The tangent estimate is 0.495081 rad, while the full expression gives 0.495069 rad. 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 = 1.5707963267948966
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 incident angle inside the stated interval, predict whether transmitted angle 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: Angles are measured from the normal and the indices are fixed at the chosen wavelength. 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 rad: 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(1.5707963267948966) - evaluate(0.0))
for points in [101, 201, 401]:
grid = np.linspace(0.0, 1.5707963267948966, 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.