Optical instruments
An optical instrument guides light from an object to an eye, film, sensor, or intermediate image. Cameras, eyes, magnifiers, microscopes, and telescopes serve different purposes, but all combine image formation, aperture, wavelength, detector response, and a focus condition.
Magnification alone does not guarantee added detail. A larger displayed image may only enlarge blur that was already set by diffraction, aberration, atmospheric turbulence, motion, focus error, or detector sampling.
Imaging foundation
For a thin lens under a real-is-positive sign convention,
\[\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}\]and transverse magnification is
\[m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}\]The image from one element can become the object for the next. Its signed location must be converted into the next element's coordinate convention. Collapsing a multi-element instrument into one unexplained “zoom” value hides focus, intermediate images, and pupil locations.
The eye
The cornea and crystalline lens focus light onto the retina. The eye changes optical power through accommodation rather than moving the retina. A relaxed normal eye focuses distant objects on the retina; a near object requires greater converging power.
The near point is the closest distance at which the eye can maintain clear focus. A conventional reference value for textbook angular magnification is
\[D=25\ \mathrm{cm}\]This is a standard comparison distance, not a universal biological constant. Actual near points vary with individual and age.
Myopia focuses distant light in front of the retina and is corrected with a diverging lens. Hyperopia tends to focus near-object light behind the retina and is corrected with a converging lens. A prescription in diopters uses
\[P=\frac{1}{f}\]with focal length in metres.
Camera
A camera lens forms a real image on a sensor. Focusing changes the relation among object, lens, and sensor so the sensor lies at the image plane. For a very distant object, \(d_o\to\infty\) and \(d_i\approx f\).
The f-number is
\[N=\frac{f}{D_{\mathrm{aperture}}}\]A smaller f-number means a larger entrance aperture for the same focal length. It admits more light and reduces diffraction blur but usually decreases depth of field and may expose more lens aberration. Exposure also depends on shutter time and sensor sensitivity; f-number is not a direct measure of final image brightness without those settings.
For a circular aperture, the diffraction-limited Airy radius in the image plane is
\[r_1\approx1.22\frac{\lambda f}{D_{\mathrm{aperture}}} =1.22\lambda N\]Closing the aperture increases this diffraction scale even if geometric depth of field improves.
Simple magnifier
A converging lens used with the object inside its focal length produces an upright virtual image. It increases angular size at the eye. For a relaxed eye viewing an image at infinity, approximate angular magnification is
\[M\approx\frac{D}{f}\]If the final virtual image is placed at the reference near point, the common approximation is
\[M\approx1+\frac{D}{f}\]These are angular magnifications, not transverse image-height ratios. They compare the viewing angle with and without the instrument under specified eye conditions.
Compound microscope
A compound microscope uses a short-focal-length objective to form an enlarged real intermediate image. An eyepiece then acts as a magnifier for that image. Under a common long-tube approximation with relaxed final viewing,
\[|M|\approx \frac{L}{f_o}\frac{D}{f_e}\]Here \(L\) is an effective tube length, \(f_o\) is the objective focal length, and \(f_e\) is the eyepiece focal length. Sign conventions determine inversion; the magnitude describes angular enlargement.
Resolution depends strongly on wavelength and numerical aperture, not merely magnification. A common lateral resolution scale is
\[d\approx0.61\frac{\lambda}{\mathrm{NA}}\]with
\[\mathrm{NA}=n\sin\alpha\]where \(n\) is the medium index and \(\alpha\) is the accepted cone half-angle. Empty magnification beyond detector or eye resolution adds size without revealing new spatial information.
Astronomical telescope
A telescope objective collects light and forms an image near its focal plane. The eyepiece increases the angular size seen by the eye. For an afocal telescope adjusted for a relaxed eye, angular magnification is
\[M=-\frac{f_o}{f_e}\]The negative sign for two converging lenses indicates an inverted image. A Galilean telescope uses a diverging eyepiece and gives an upright image with different length and field-of-view behavior.
The objective diameter controls ideal diffraction-limited angular resolution:
\[\theta_R\approx1.22\frac{\lambda}{D_o}\]A larger focal length makes a larger image scale at the focal plane, but a larger aperture is what narrows the ideal Airy pattern. Ground-based observations may instead be limited by atmospheric seeing unless adaptive optics or other methods are used.
Afocal system check
Two thin lenses separated by the sum of their positive focal lengths form an ideal Keplerian afocal telescope. Parallel rays enter and parallel rays leave, so the final image is at infinity for a relaxed eye.
import numpy as np
def lens(f):
return np.array([[1.0, 0.0], [-1 / f, 1.0]])
def propagation(distance):
return np.array([[1.0, distance], [0.0, 1.0]])
objective, eyepiece = 0.5, 0.05
system = lens(eyepiece) @ propagation(objective + eyepiece) @ lens(objective)
for height in [-0.01, 0.0, 0.01]:
outgoing = system @ np.array([height, 0.01])
print("Output height and paraxial angle:", outgoing)
assert abs(system[1, 0]) < 1e-12 # Output angle does not depend on input height.
If the separation changes, the outgoing rays converge or diverge and the final image is not at infinity.
Paraxial matrices
In first-order optics, a ray can be represented by height and small angle. Propagation through distance \(d\) and a thin lens can be represented by matrices:
\[\begin{pmatrix}y_2\\\theta_2\end{pmatrix} =\begin{pmatrix}1&d\\0&1\end{pmatrix} \begin{pmatrix}y_1\\\theta_1\end{pmatrix}\] \[\begin{pmatrix}y_2\\\theta_2\end{pmatrix} =\begin{pmatrix}1&0\\-1/f&1\end{pmatrix} \begin{pmatrix}y_1\\\theta_1\end{pmatrix}\]Multiplying matrices in physical order models a sequence of elements. Matrix order is not commutative; reversing two elements generally describes a different instrument.
Detector and sampling limits
A sensor records pixels rather than a continuous ideal image. Pixel pitch, color filters, fill factor, noise, dynamic range, exposure, and processing affect the result. Sampling too coarsely can alias fine optical detail. Sampling very finely does not overcome a blur already imposed by diffraction or aberration.
The eye likewise has finite receptor spacing, neural processing, contrast sensitivity, and an adjustable pupil. An optical calculation should name its final observer or detector instead of reporting magnification as if all receivers were identical.
Verification and limits
A one-element pipeline must reproduce the thin-lens equation. Moving a real sensor to the calculated image distance should minimize paraxial defocus. An afocal telescope should map parallel input to parallel output. Doubling circular aperture diameter should halve the ideal Rayleigh angular limit at fixed wavelength.
First-order models omit lens thickness, aberrations, coatings, vignetting, scattering, polarization, chromatic dispersion, diffraction details, mechanical tolerances, motion, atmospheric turbulence, detector noise, and image processing. The Rayleigh criterion and near-point conventions are useful reference models, not guarantees of real performance.
A worked Python notebook
Combine an objective of focal length 0.5 metres with a positive eyepiece in an ideal afocal telescope. The question for this calculation is precise: how does angular magnification change when we change eyepiece focal length, 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 minus sign indicates inversion in the paraxial model. 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 0.02 to 0.1 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 eyepiece focal length, expressed in m. The output is
angular magnification, expressed in dimensionless. 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 0.05 m, the expression gives -10 dimensionless. 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/q", locals={"q": q})
reference_input = 0.05
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted angular magnification:", reference_output, "dimensionless")
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 -25 dimensionless. The midpoint, at 0.06 m, gives -8.33333 dimensionless. The upper endpoint gives -5 dimensionless. Read these together with the assumptions: they describe different choices of eyepiece focal length 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([0.02, 0.060000000000000005, 0.1])
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} dimensionless")
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 eyepiece focal length; the vertical axis shows angular magnification. 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(0.02, 0.1, 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("Eyepiece focal length (m)")
ax.set_ylabel("Angular magnification (dimensionless)")
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 angular magnification change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.05 m is 200 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.
\[f'(q) = \frac{0.5}{q^{2}}\]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 = 0.0504
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, "dimensionless")
print("Full expression:", full_prediction, "dimensionless")
print("Approximation difference:", full_prediction - linear_prediction, "dimensionless")
Here the changed input is 0.0504 m. The tangent estimate is -9.92 dimensionless, while the full expression gives -9.92063 dimensionless. 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 = 0.08
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 eyepiece focal length inside the stated interval, predict whether angular magnification 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 minus sign indicates inversion in the paraxial model. 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 dimensionless: 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(0.1) - evaluate(0.02))
for points in [101, 201, 401]:
grid = np.linspace(0.02, 0.1, 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.