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Achromatic doublet power balance chooses two signed optical powers to meet two goals: obtain the required focal length and cancel the primary F–C colour difference. Model two thin lens elements in contact in air, using the paraxial approximation. Let $i=c,f$ label the crown glass and flint glass elements, and let ${\Phi }_{i,d}$ denote each element's optical power at the reference d line. The Abbe number for each material isThe subscripts label standard spectral lines: blue F near $486.1\mathrm{nm}$, red C near $656.3\mathrm{nm}$, and reference d near $587.6\mathrm{nm}$. These are three specified wavelengths, not colour bands. A larger Abbe number means less relative optical dispersion between F and C. ^[ref].

For a fixed thin lens shape in air, write the lensmaker's equation as ${\Phi }_i(\lambda )=K_i[n_i(\lambda )-1]$, with signed shape factor $K_i=1/R_{i,a}-1/R_{i,b}$. Subtracting its C value from its F value gives the primary optical power differenceThe negative element has negative $\Delta {\Phi }_i$: its blue optical power is more negative, not less negative. The contact rule for optical power therefore givesThis zero is equality of the total F and C optical powers, hence equality of their focal lengths when the total optical power is nonzero. The relation follows directly from the lensmaker's equation and the Abbe number; it is exact within the fixed-shape contact thin lens model. ^[ref]. Tatum uses a yellow D-line reference; here the Abbe number convention uses the d line consistently.

For a toy design, choose $V_c=60$, $V_f=30$, ${\Phi }_{c,d}=10\mathrm{D}$ and ${\Phi }_{f,d}=-5\mathrm{D}$. ThenThe positive element contributes $+1/6\mathrm{D}$ to the F–C difference and the negative element contributes $-1/6\mathrm{D}$. This retains the same reference focal length as the single $5\mathrm{D}$, $V_d=60$ thin lens in chromatic aberration, while cancelling its primary colour split. The values are illustrative design parameters, not a catalogue optical glass pair.

Use the same synthetic colour coordinate $q=-1/2,0,+1/2$ and toy optical dispersion as in chromatic aberration. The two element optical powers and their sum becomeAt the red endpoint, the element optical powers are $9.916667\mathrm{D}$ and $-4.916667\mathrm{D}$; at the blue endpoint, they are $10.083333\mathrm{D}$ and $-5.083333\mathrm{D}$. Both sums are $5\mathrm{D}$. In this first-order toy model alone, the sum is constant for every $q$ and the focal length remains $0.2\mathrm{m}$ throughout. Real material optical dispersion is not captured by these two straight slopes, so the flat line is not a claim that a real achromatic doublet focuses all colours perfectly.

For specified nonzero reference optical power ${\Phi }_d$ and distinct Abbe numbers, solving the two balance equations givesThus ${\Phi }_d>0$ and $V_c>V_f>0$ require positive crown glass optical power and negative flint glass optical power. If the two finite Abbe numbers were equal, cancellation would force zero total optical power. If they are close, the required element optical powers become large compared with the total: cancelling primary colour can demand much stronger individual elements.

The contact assumption matters. For two thin lens elements separated by distance $s$ in air, the equivalent optical power is ${\Phi }_c+{\Phi }_f-s{\Phi }_c{\Phi }_f$, evaluated at each wavelength; its colour change need not satisfy the contact balance. Finite thickness, departures from the paraxial approximation, and the secondary spectrum also limit the simple prediction.^[ref]

Axial chromatic aberration, also called longitudinal chromatic aberration, is colour-dependent focus position along the optical axis. For incident light rays parallel to the optical axis, it appears as a changing focal length. The chromatic aberration diagram isolates this effect using an ideal thin lens at each colour.

KIT's teaching apparatus uses a glass spoke-wheel or test negative as the object, a halogen lamp with an illuminating optical lens, an imaging optical lens, a screen and a triangular rail with a position scale. The illuminating optical lens concentrates the lamp filament's image on the imaging optical lens's centre; translucent paper checks that illumination. This reduces the contribution of peripheral light rays and their spherical aberration. The instructions require fixed object-screen separation exceeding $4f$, at least ten displacement measurements, and separate red- and blue-filter measurements to test chromatic aberration.^[ref]

The following measurement protocol elaborates that apparatus rather than reporting additional KIT results. Align the target centre, optical lens and screen normal along one optical axis; measure $L$ between the target pattern plane and receiving screen face. Lock the target and screen. Slide the imaging optical lens to obtain the enlarged real image, then the reduced real image, reading the same carrier reference mark at each setting. Record $r_{-},r_{+}$ and $d=r_{+}-r_{-}$ for each pair. Record corresponding target and image intervals to test the reciprocal transverse magnifications independently; merely observing different image sizes is a weaker test.

Use the same central target detail to judge best optical focus. Bracket each setting by moving through both visibly blurred sides, repeat from both travel directions, and record the accepted position range as well as repeat scatter. Rail divisions set a readout limit but do not alone specify focusing uncertainty. Check scale calibration, carrier play and drift; a common additive carrier offset cancels in $d$, whereas scale-factor error remains. Keep illumination, aperture, target, screen and optical lens orientation fixed within a series. Change the aperture and colour filter in separate control series: spherical aberration and chromatic aberration can shift the apparent best optical focus, while a very small aperture introduces stronger diffraction and makes focus discrimination harder. Record these settings alongside the inferred focal length, rather than interpreting all disagreement as random measurement error.

For a rigid thick lens in air, let $h=x_{H^{\prime }}-x_H$ be the signed separation of its rear and front principal planes along the direction of propagation. Then the positive object and image distances measured from the principal planes obey $u+v=L-h$. Translating the whole assembly without reversing it moves both principal planes equally, so the measured carrier displacement still equals the difference of the two solutions for $u$. The same algebra therefore givesHere $f$ is the effective focal length measured from the principal planes. This deduction uses the principal-plane imaging equation, not the uncorrected thin lens formula.^[ref] Mechanical centre thickness is not generally $h$. Record thickness and optical lens orientation, but use a justified bound or model for $h$ if it matters at the claimed precision. The two readings $L,d$ alone cannot determine both $f$ and unknown $h$; cancelling a carrier offset does not cancel this limitation.

An optical camera can focus a white object sharply in one colour while leaving coloured fringes around its edges in another. Chromatic aberration is the dependence of image position or image scale on wavelength, caused by optical dispersion in the optical lens materials. Axial chromatic aberration separates the optical foci along the optical axis; lateral chromatic aberration changes the transverse magnification with wavelength. Moving the receiving plane can favour one colour without bringing all colours into focus. ^[ref]; ^[ref].

For ordinary optical glass in the visible range, the blue refractive index exceeds the red refractive index. For a fixed positive thin lens in air, the lensmaker's equation then givesHere $\lambda$ is wavelength, $n$ is refractive index, $\Phi$ is optical power, $f$ is focal length, and $R_a,R_b$ are the signed surface radius of curvature values in the lensmaker's equation. The blue light rays therefore bend more strongly and meet sooner. This ordering is a statement about ordinary visible optical dispersion, not a universal rule for every material and every spectral range. ^[ref].

For a worked example, take a single positive thin lens with reference optical power ${\Phi }_d=5\mathrm{D}$ and Abbe number $V_d=60$, hence reference focal length $f_d=0.2\mathrm{m}$. Its F–C optical power difference is $\Delta \Phi ={\Phi }_F-{\Phi }_C={\Phi }_d/V_d=5/60\mathrm{D}\approx 0.08333\mathrm{D}$, by the achromatic doublet power balance relation below. This difference alone does not locate the reference optical power within the F–C interval. To draw a transparent example, introduce a synthetic colour coordinate $q$, a dimensionless quantity, with red $q=-1/2$, reference $q=0$, and blue $q=+1/2$, and choose the toy optical dispersion modelThe diagram's red and blue endpoints represent the C and F endpoints of this toy model. Its reference point is placed symmetrically in optical power by construction; $q$ is not a measured wavelength scale, and the model is not measured optical glass data. The endpoint focal lengths areThus the axial chromatic aberration in this example separates the endpoint optical foci by about $3.334\mathrm{mm}$. A single receiving plane cannot coincide with both optical foci.

The useful design idea is to cancel changes of optical power while retaining a positive total optical power. An achromatic lens does this with materials having different Abbe numbers. It greatly reduces chromatic aberration, but correcting two chosen wavelengths still leaves the possibility of a secondary spectrum.

A diffraction-limited system has an optical response close to the ideal diffraction response for its specified optical pupil, wavelength and optical focus. Improving the optical aberrations further makes little difference to that response. This describes the optics: a poorly sampled or noisy detector can still deliver an image with much worse usable diffraction-limited resolution.

The optical pupil determines which wave amplitudes and wave phases contribute to an image point. In a scalar, paraxial approximation at the focal plane of an optical lens in air, write the pupil transmission as $T(u,v)e^{i\varphi (u,v)}$, after removing the ideal focusing phase. The amplitude spread function is proportional to its scaled Fourier transform:Here $(u,v)$ are pupil coordinates, $(x,y)$ are image coordinates, $f$ is focal length, and $\lambda$ is the wavelength in air. For a uniformly transmitting, unobscured circular optical pupil at ideal optical focus, the result is an Airy disk with surrounding rings. A blocked centre, support structures, or nonuniform transmission changes the ideal point spread function; “diffraction limited” does not specify one universal spot shape. The existing Fourier model of a Lyot coronagraph illustrates how pupil and focal masks deliberately alter this response.^[ref]

Optical aberrations add pupil wave phase errors. Defocus contributes a quadratic pupil wave phase in the paraxial approximation, redistributing light intensity from the best-focused spot. The Strehl ratio compares the peak with the ideal response for the same pupil and total radiant flux. A broad optical spectrum produces a weighted sum of wavelength-dependent point spread functions; chromatic aberration can additionally change optical focus with wavelength. Thus the operating conditions belong with any claim that a system is diffraction limited.

Optical dispersion is the dependence of refractive index or phase velocity on frequency. It gives an optical prism wavelength-dependent deviation and causes chromatic aberration.

Electron microscope resolution concerns distinguishable specimen detail under a specified contrast criterion. It depends on diffraction, lens defects, electron energy spread, specimen motion and detection. A smaller de Broglie wavelength permits finer detail in principle, but neither wavelength nor magnification is the instrument's measured resolution.

For an electron accelerated from negligible initial kinetic energy through a positive electric potential difference $U$, combine $K=eU$, the relativistic energy-momentum relation and the de Broglie wavelength:Here $h$ is the Planck constant, $m_e$ the electron mass, $e$ the elementary charge, and $c$ the speed of light. This includes the relativistic correction omitted by the low-voltage electron accelerating-voltage wavelength approximation. With $m_e\simeq 9.1093837\times 10^{-31}\mathrm{kg}$ and SI defining values of $h,e,c$, calculated wavelengths are approximately $6.979$, $3.701$ and $1.969\mathrm{pm}$ at $30$, $100$ and $300\mathrm{kV}$ respectively. The rounded mass agrees with the recommended value.^[ref]

For a circular aperture admitting a small semi-angle $\alpha$, the Rayleigh criterion suggests a diffraction-limited resolution scale $d_D=0.61\lambda /\alpha$. At $100\mathrm{kV}$ and $\alpha =10\mathrm{mrad}$, this gives $0.226\mathrm{nm}$, already much larger than the $0.00370\mathrm{nm}$ wavelength. This is an ideal aperture estimate. Transmission electron microscopy with coherent illumination and a particular specimen requires its own image-contrast analysis; this scale alone is not a universal two-feature resolution test.

An original illustrative blur model adds competing effects:Define $C_s{\alpha }^3$ as a characteristic transverse blur from the electron-lens analogue of spherical aberration, and $C_c\alpha \Delta K/K$ as one from energy-dependent focusing, the analogue of chromatic aberration. The coefficients $C_s,C_c$ have units of length, $\Delta K$ is a chosen characteristic kinetic energy spread, and $b_0$ is a fixed residual blur. Adding their squares is a declared heuristic, not a measured point spread function or a rigorous resolution theorem. The model omits contrast transfer, specimen scattering and signal statistics, so its minimum is only an estimate within its assumptions.

Choose $C_s=C_c=1\mathrm{mm}$, $\Delta K=0.7\mathrm{eV}$ and $b_0=0.10\mathrm{nm}$ at $K=100\mathrm{keV}$. At $5\mathrm{mrad}$, the four contributions are about $0.452$, $0.125$, $0.035$ and $0.100\mathrm{nm}$, giving $b\simeq 0.480\mathrm{nm}$. At $10\mathrm{mrad}$, the diffraction contribution falls to $0.226\mathrm{nm}$ but the assumed spherical aberration contribution grows to $1.00\mathrm{nm}$. Thus opening the aperture indefinitely worsens this model's image. All these values are calculations with chosen parameters, not observations of any microscope.

Lateral chromatic aberration, also called transverse chromatic aberration, is colour-dependent image scale: an object away from the optical axis can have different image heights at different wavelengths. Its changing transverse magnification produces displaced coloured edges. The on-axis parallel bundle in the chromatic aberration diagram illustrates axial chromatic aberration, and does not measure lateral chromatic aberration. ^[ref].

The institution-issued AY 105 Lab Experiment 3: Optical aberrations by California Institute of Technology (Spring 2011 offering, Part A, pp. 2–3, 7–8) specifies a concrete optical bench: a lamp, opal diffuser and condensing optical lens illuminate a $25$ or $30\mu \mathrm{m}$ optical pinhole; a photographic telephoto optical lens acts as a collimator, producing an approximately $75\mathrm{mm}$ beam. A test optical lens sits on a rotation stage, followed by an adjustable optical microscope. The procedure establishes the on-axis image, then rotates the test optical lens a few degrees and examines the off-axis image through optical focus. It compares this readout with through-focus ray tracing at the corresponding field angles and offers a Ronchi ruling to calibrate the measuring eyepiece.

Part B (pp. 9–10) substitutes a spherical mirror of $1\mathrm{m}$ radius of curvature, with the optical microscope beside the incoming beam so the image is necessarily off-axis. It calls for measuring the field angle, calibrating the eyepiece and recording the image's length and width at minimum size. These are specified apparatus dimensions and instructions, not supplied measurement results or dimensions of the normalized figures.

A controlled extension would record images at zero and equal positive and negative field angles, repeat zero afterwards, and scan through optical focus at every setting. Keep the illuminated optical pupil and illumination fixed, recenter the optical microscope without clipping the bundle, and use its calibrated scale to separate a mere image displacement from a change in blur shape. The pure Seidel coma pupil mapping predicts reversal and linear growth; a stationary asymmetric central image instead calls for checking alignment. A focus-dependent change between perpendicular elongations suggests an astigmatism contribution. These comparisons test the model rather than assuming every off-axis flare is pure coma.

The optical pinhole has finite angular size, approximately its diameter divided by the collimator's focal length; the collimator and observing optical microscope also contribute their own optical aberrations. A broadband lamp can mix chromatic aberration with the geometric blur, and diffraction limits the smallest visible structure. A narrow spectral band, repeated field settings and a reference image help separate these effects. Consequently a measured length and width are useful checks on off-axis image formation, but they cannot by themselves determine a unique coma coefficient or recover the full point spread function.

Atomic-resolution cryo-electron microscopy became possible by retaining high-frequency structural information through the whole experiment: specimen preparation, electron optics, detection and computation. The decisive advances improved the information obtained per damaging electron and made it possible to align and average many weak images without averaging away structural differences. A short de Broglie wavelength had always been available; the practical limit was preserving and extracting the information carried by those electrons.

First, vitrification preserves hydrated biological material in a thin noncrystalline layer of water. Rapid freezing avoids the strong periodic scattering and disruption associated with ordinary ice crystals, while cryogenic imaging slows some damage processes. The specimen still suffers radiation damage, and a thicker layer adds background and multiple scattering. Reproducible thin ice, clean grids, stable supports and reduced mechanical drift are therefore prerequisites. The specimen must remain structurally intact and occupy enough orientations for a three-dimensional reconstruction. Adsorption at the air–water interface, preferred orientation and variable thickness can defeat an otherwise excellent microscope. Sample optimization should include independent checks of assembly integrity and activity where feasible.

Second, direct electron detection greatly improved the useful signal relative to earlier indirect cameras. In a counting detector, individual electron events can be localized and recorded without treating the deposited signal merely as a continuous intensity. detective quantum efficiency measures how efficiently the detector preserves the input signal-to-noise ratio as a function of spatial frequency. High efficiency matters particularly because the permissible exposure is limited. At excessive event rates, overlapping events cause coincidence losses, so faster readout permits useful counting at practical collection rates. A nominally smaller camera pixel alone cannot replace a detector that preserves high-frequency information.

Fast detectors also turn an exposure into a sequence of movie frames. Aligning those frames corrects specimen drift and beam-induced movement; local correction handles different movements across the imaged area. Uncorrected movement preferentially smears fine features. dose weighting then combines early and later frames according to the information they retain: later exposure can help low-frequency alignment even after fine structural details have been damaged. ^[ref] exemplifies this computational use of detector movies. Frame alignment cannot restore chemistry already destroyed by radiation damage, and extremely noisy individual frames require constrained motion estimates.

Third, stable and coherent electron optics increase contrast transfer at the finest spacings. A narrow source energy spread reduces chromatic aberration, because electrons with different energies otherwise focus differently. A cold field emission gun provides one route; a monochromator provides another. An energy filter removes many inelastically scattered electrons that otherwise contribute background. This does not remove all damage or every unwanted scattering event. Improved mechanical and electrical stability, reproducible lens settings, accurate coma-free alignment and calibrated magnification make the information more consistent across images.

The 2020 apoferritin reconstructions illustrate the final hardware step. One study combined a cold source, improved energy filtering and a fast direct camera with advanced correction to obtain a map at about $1.22\text{A˚}$. Another used an instrument with monochromation and aberration correction to reach about $1.25\text{A˚}$.^[ref][ref] The two instruments demonstrate complementary ways of preserving coherent high-frequency information. These unusually rigid, symmetric particles are favorable test specimens; their resolutions do not imply that a flexible cellular assembly will reach the same limit.

Fourth, modern reconstruction treats the noisy images statistically. single-particle analysis estimates how many randomly oriented projections correspond to a common three-dimensional structure. Bayesian inference and maximum likelihood estimation permit uncertainty in image orientations and class membership to be handled rather than committing every weak image to a single exact alignment from the beginning. ^[ref] provides a concrete implementation. Two-dimensional classes reject many damaged or overlapping particles; three-dimensional classification can separate different compositions or conformations. Appropriate symmetry increases the effective information, but an incorrect symmetry assumption can erase a real asymmetric feature.

Refinement increasingly corrects individual particle movements and optical imperfections. Per-particle defocus estimates accommodate height differences in ice. Beam tilt, higher-order aberrations and anisotropic magnification need correction when their accumulated phase errors become important. The contrast transfer function changes the magnitude and sign of Fourier components; combining images recorded with different defocus values recovers information suppressed at particular transfer-function zeros. Near atomic spacings, accounting for Ewald sphere curvature can further improve reconstruction. Such corrections require adequate signal and calibration; simply adding more adjustable parameters to weak data risks fitting noise.

Automation and fast computation make these methods practical. Stable automated acquisition and beam-image-shift strategies collect large image sets; parallel computation permits repeated alignment, classification and refinement. More particles improve the average only if they carry consistent signal. Under an ideal independent-noise model, the amplitude signal-to-noise ratio improves in proportion to $\sqrt{N}$; a systematic alignment error or persistent structural heterogeneity will not disappear by this scaling. For a target spacing $d$ and real-space pixel size $a$, Nyquist–Shannon sampling theorem requires $a\le d/2$. For $d=1.2\text{A˚}$ this gives $a\le 0.6\text{A˚}$ per pixel, a necessary sampling condition rather than a guarantee of such resolution.

Validation is part of the technological achievement. Split particles into independent halves and refine them independently; calculate their Fourier shell correlation, commonly reporting the spacing at the $0.143$ criterion for independently refined half-maps. Shared high-resolution references, poorly handled masks or correlated data can inflate agreement, so independence and mask correction matter. Inspect directional coverage and local resolution, compare models against held-out map information, and check stereochemistry. The global resolution is a summary: flexible loops, bound lipids and minor components may be much worse than the ordered core.

Finally, distinguish an atomistic model from atomically resolved data. At ordinary near-atomic resolution, side chains and backbone features can support building an atomic model while individual neighboring chemical atoms are not all separated. At true atomic resolution, many individual atomic peaks become distinguishable and some hydrogen contributions may be detectable under favorable conditions. cryo-electron microscopy reconstructs scattering electrostatic potential, so interpretation of very fine features is not identical to interpreting X-ray crystallography electron-density maps. The breakthrough was the combined improvement of specimen quality, counting movies, optical coherence and statistical reconstruction, with independent validation showing that the fine detail was supported by the data.

Transmission electron microscopy accelerates electrons through a thin sample, then uses electron lenses to form an image or diffraction pattern. Elastic scattering encodes projected structure and potential; inelastic scattering carries excitation and chemical information. Thin specimens reduce unwanted multiple scattering and absorption, although coherent dynamical scattering can still matter in a crystal. High-resolution phase-contrast images depend on defocus and the transfer function as well as the specimen; they are not direct height maps.

The relativistic wavelength is $\lambda =h/\sqrt{2m_{e}eV[1+eV/(2m_e{c}^2)]}$. A shorter wavelength permits fine resolution but does not guarantee it. A schematic probe/image blur budget contains diffraction $\sim \lambda /\alpha$, spherical aberration $\sim C_s{\alpha }^3$ and chromatic aberration $\sim C_c\alpha \Delta E/E$, plus source size, instabilities, specimen motion and radiation damage. Optimizing the aperture angle balances these terms; increasing accelerating voltage can also alter damage and contrast.

Bright-field imaging accepts the central transmitted beam or its neighborhood. Regions that scatter electrons outside that acceptance can appear dark. Conventional dark-field imaging selects a diffracted beam. Annular dark-field electron imaging in a scanning TEM instead collects a range of scattered angles while a probe is scanned. A sufficiently high-angle detector often gives strong atomic-number sensitivity and more nearly incoherent contrast; thickness, channeling and detector geometry prevent a universal conversion of intensity to composition.

Electron energy loss spectroscopy measures energy lost by transmitted electrons. Core-ionization thresholds identify elements, while near-edge fine structure probes bonding and unoccupied states. At successive probe positions, one can record a spectrum image or form a map from a selected edge after background subtraction. Plural scattering, edge overlap, delocalization, specimen thickness and counting statistics limit the interpretation. Chemical specificity comes from excitation energies and spectral structure, not from brightness alone.

A development already established before this 2012 paper is aberration-corrected transmission electron microscopy: multipole correctors compensate leading lens aberrations, allowing useful larger angular acceptance and finer probes. Correction does not remove every chromatic, stability or specimen limit. A concrete application was atom-by-atom annular-dark-field analysis of monolayer boron nitride with substitutional defects, resolving chemical differences between sites in a nonuniform lattice.^[ref] Combining structural maps with spatially resolved core-loss spectra supplies another route to distinguish composition and bonding at interfaces. Both exploit a calibrated imaging or spectroscopy response rather than assuming every visible spot is chemically self-identifying.

For thin contacting components, optical powers add. At wavelength $\lambda$ the total optical power is $\Phi (\lambda )={\rho }_1[n_1(\lambda )-1]+{\rho }_2[n_2(\lambda )-1]$. An achromatic lens has ${\Phi }_F={\Phi }_C$, henceFor normally dispersive optical glasses, both index differences are positive, so the two components have opposite curvature factors and opposite optical powers. This uses the thin Lensmaker's equation; finite thickness and separated principal planes require a more detailed design. If the denominator vanishes, use the undivided power-balance equation rather than this ratio. Two-wavelength correction alone does not eliminate the residual chromatic aberration at every intermediate wavelength.

Set the red image at coordinate zero, the blue image at $\delta =0.010\mathrm{m}$, and the screen at $D=1.000\mathrm{m}$. Let the correcting thin lens be at $x$, with $\delta <x<D$. Both original images are then real objects for the correcting thin lens, with positive object distances $u_r=x$, $u_b=x-\delta$, and common positive image distance $v=D-x$. The thin lens equation givesThe refractive indices and Lensmaker's equation requireSubstitute the two focal lengths and cancel $D-x$:Thus the correcting thin lens is $100\mathrm{mm}$ beyond the red image and $900\mathrm{mm}$ before the screen. The required focal lengths and surface radius areThe blue focal length is shorter because the blue refractive index is larger; these two differing object distances are mapped to the same screen, removing the specified longitudinal chromatic aberration.

A real image lies where the outgoing light rays from each object point actually intersect in geometrical optics. A screen or detector placed at that image plane receives the focused pattern. The screen reveals the real image; it does not create the intersection. A viewer can also see a real image without a screen if their eye receives the diverging light rays beyond the intersection. ^[ref].

For a converging thin lens and real object with $u>f>0$, the real image has $v>0$ and signed transverse magnification $m<0$. Inversion is a property of this particular setup, not the definition of a real image. With $f=10\mathrm{cm}$ and $h_o=0.50\mathrm{cm}$, the following are predictions:

To test both image position and transverse magnification, use an optical bench with an illuminated arrow or millimetre target, a converging optical lens of nominal focal length $10\mathrm{cm}$, a movable white screen and a ruler. Measure corresponding target and image intervals, rather than guessing the whole image size when it exceeds the screen. PASCO's Experiment 12 compares distance-based and size-based magnification; the following procedure uses several object distances, as in the Northern Virginia laboratory. ^[ref]; ^[ref].

Keep the optical lens, illumination and central target detail fixed between trials. Approach best focus from both sides and record the interval over which sharpness is difficult to distinguish; repeat focusing to estimate its reproducibility. A modest central aperture stop tests whether peripheral light rays change the result, but also reduces brightness. Spherical aberration, chromatic aberration and diffraction can broaden the optimum; a thick optical lens also makes the chosen reference plane imperfect. Treat nominal $f$ as an initial prediction, and report position and interval-reading limits instead of forcing exact agreement. ^[ref].

Moving the object toward the front optical focus from outside makes the real image larger and farther away. A failed screen search near $u=f$ can therefore mean that the real image is beyond the bench, rather than that a virtual image has formed. At exactly $u=f$, the outgoing bundle from each point is parallel and no finite screen position focuses it.

The secondary spectrum is residual chromatic aberration after an achromatic lens brings two selected wavelengths to the same optical focus. For example, equal red and blue focal lengths can coexist with a different green focal length. The Abbe number fixes one F–C relative difference, not the complete shape of optical dispersion between or beyond those wavelengths. The perfectly flat sum in the achromatic doublet power balance diagram deliberately omits this residual effect. ^[ref].