A Byte of PhysicsLogo

Mirrors and lenses

Mirrors and lenses redirect light to form images. Ray optics replaces a wavefront with lines perpendicular to it, then applies reflection or refraction at each surface. The model works well when optical elements and apertures are large compared with wavelength and when rays stay close enough to the optical axis for the paraxial approximation.

An image is real when physical outgoing rays converge at it and can be projected onto a screen. An image is virtual when outgoing rays diverge but their backward extensions meet. A negative image distance is therefore meaningful, not a failed calculation.

Plane mirrors

For a plane mirror, reflected rays appear to originate from the same distance behind the mirror as the object lies in front. The image is virtual, upright, and equal in size. Using the real-is-positive convention for image distance,

\[d_i=-d_o,\qquad m=+1\]

The image reverses front-to-back relative to the object; the familiar apparent left-right reversal follows from how observers compare themselves with the image.

Spherical mirrors

A concave spherical mirror converges paraxial rays and has positive focal length under the real-is-positive convention. A convex mirror diverges rays and has negative focal length. For a spherical mirror of radius of curvature \(R\),

\[f=\frac{R}{2}\]

The mirror equation is

\[\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}\]

and transverse magnification is

\[m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}\]

A negative magnification means the image is inverted relative to the object; a positive one means upright. Its magnitude is the ratio of image height to object height.

Thin lenses

For a thin lens in air under the same real-is-positive convention,

\[\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}\]

A converging lens has positive focal length and a diverging lens has negative focal length. The magnification relation remains

\[m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}\]

Although the mirror and thin-lens equations look identical in this convention, their ray rules and physical surfaces differ. References using a Cartesian convention may assign different signs; convert the entire system rather than borrowing one sign from each convention.

Solving with an explicit focal-plane branch

Rearrange the imaging equation as

\[\frac{1}{d_i}=\frac{1}{f}-\frac{1}{d_o}\]

If the right side is zero, outgoing rays are parallel and the image is at infinity.

import numpy as np


def thin_lens(focal_length, object_distance):
    inverse_distance = 1 / focal_length - 1 / object_distance
    if abs(inverse_distance) < 1e-12:
        return np.inf, None
    image_distance = 1 / inverse_distance
    return image_distance, -image_distance / object_distance


for distance in [0.3, 0.1, 0.05]:
    print(
        "Object distance (m):",
        distance,
        "image distance and magnification:",
        thin_lens(0.1, distance),
    )

The near-zero tolerance should be scaled to the input values and precision. It represents numerical classification near an ideal singular limit, not a finite physical image.

Principal rays for a converging lens

Three convenient paraxial rays locate the image:

  1. A ray parallel to the optical axis refracts through the far focal point.
  2. A ray through the near focal point emerges parallel to the axis.
  3. A ray through the optical center continues approximately straight for a thin lens.

Any two rays locate the ideal image; the third checks the construction. For a diverging lens, the parallel incident ray emerges as if it came from the near focal point, and the other ray rules are adjusted by reversing the appropriate extensions.

Principal rays are selected examples from all rays leaving an object point. Blocking one does not normally remove one part of the image; it reduces brightness and can change diffraction and aberration.

Trace the two blue rays in Figure 1 from the object tip. The horizontal incident ray bends through the far focal point, while the other continues through the lens center. Their intersection is below the axis on the far side, so the image is inverted and real. Measuring from the lens places both object and image at twice the focal distance.

Converging-lens image regions

Use Figure 1 as the reference case: equal object and image distances give magnification minus one. Moving the object changes where the two rays meet, producing the cases below. Let the object distance be positive:

A diverging thin lens with a real object produces a virtual, upright, reduced image under ordinary conditions.

Worked lens example

Use a converging lens with \(f=10\ \mathrm{cm}\) and an object at \(d_o=30\ \mathrm{cm}\). Then

\[\frac{1}{d_i} =\frac{1}{10\ \mathrm{cm}} -\frac{1}{30\ \mathrm{cm}} =\frac{1}{15\ \mathrm{cm}}\]

so

\[d_i=15\ \mathrm{cm},\qquad m=-\frac{15}{30}=-0.50\]

The positive image distance means real; the negative magnification means inverted; the magnitude one-half means half as tall.

Move the same object to five centimetres from the lens:

\[\frac{1}{d_i} =\frac{1}{10\ \mathrm{cm}} -\frac{1}{5\ \mathrm{cm}} =-\frac{1}{10\ \mathrm{cm}}\]

Now \(d_i=-10\ \mathrm{cm}\) and \(m=+2\): the image is virtual, upright, and twice as tall.

Lensmaker's equation

For a thin lens in air with refractive index \(n\) and signed surface radii \(R_1\) and \(R_2\), one common convention gives

\[\frac{1}{f} =(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)\]

The radii signs depend on surface centers of curvature and propagation direction. The equation shows why lens material and curvature determine focal length. Because index depends on wavelength, focal length can also depend on color.

Optical power and lenses in contact

Optical power is

\[P=\frac{1}{f}\]

When focal length is in metres, power is measured in diopters. Thin lenses in contact have approximately additive powers:

\[P_{\mathrm{eq}}=P_1+P_2, \qquad \frac{1}{f_{\mathrm{eq}}} =\frac{1}{f_1}+\frac{1}{f_2}\]

Separated lenses require propagation between elements; their powers do not simply add. For a sequence, the image from one element becomes the object for the next, with distances and signs recomputed in one consistent coordinate system.

Apertures and brightness

The imaging equation locates the paraxial image but does not determine brightness or resolution. A finite aperture limits the bundle of rays, changes irradiance, and creates diffraction. Closing an aperture often increases depth of field and reduces geometric aberrations while increasing diffraction blur. No ideal point image remains a mathematical point in a real finite-wavelength system.

Aberrations and limits

Spherical surfaces focus marginal rays differently from paraxial rays, producing spherical aberration. Refractive-index dispersion produces chromatic aberration. Off-axis objects can show coma, astigmatism, field curvature, and distortion. Multi-element objectives use different materials and surfaces to reduce these errors.

The thin-element equations assume small angles, negligible thickness, homogeneous media, and ideal surfaces. They omit diffraction, interference, coatings, absorption, scattering, sensor response, and polarization effects. Verify the two-focal-length case, focal-plane limit, virtual-image branch, and ray-equation agreement before extending the model.

A worked Python notebook

Form a real image using a converging thin lens with focal length 0.1 metres. The question for this calculation is precise: how does image distance change when we change object distance, 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.

The chosen range stays beyond the focal plane; crossing it changes the image branch. 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 0.12 to 0.6 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 object distance, expressed in m. The output is image distance, expressed in 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.

\[f(q) = \frac{1}{10 - \frac{1}{q}}\]

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 0.3 m, the expression gives 0.15 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("1/(10-1/q)", locals={"q": q})
reference_input = 0.3
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted image distance:", reference_output, "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 0.6 m. The midpoint, at 0.36 m, gives 0.138462 m. The upper endpoint gives 0.12 m. Read these together with the assumptions: they describe different choices of object distance 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([0.12, 0.36, 0.6])
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} 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 object distance; the vertical axis shows image distance. 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(0.12, 0.6, 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("Object distance (m)")
ax.set_ylabel("Image distance (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 image distance change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.3 m is -0.25 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.

\[f'(q) = - \frac{1}{q^{2} \left(10 - \frac{1}{q}\right)^{2}}\]

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 = 0.3024
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, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 0.3024 m. The tangent estimate is 0.1494 m, while the full expression gives 0.149407 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 = 0.48
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 object distance inside the stated interval, predict whether image distance 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: The chosen range stays beyond the focal plane; crossing it changes the image branch. 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 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(0.6) - evaluate(0.12))
for points in [101, 201, 401]:
    grid = np.linspace(0.12, 0.6, 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.

Sources

Share to: